Limits - Thunderbird
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This article describes the known limits. However, there is always the possibility of a configuration specific bug or a bug fix for a lower limit breaking in a maintenance release.
There is no known limit of the number of accounts you can have. If you run into one, it probably would be either a bug in how many accounts the folder pane can display or due to "Use Global Inbox" being checked for that account. One user is using 150 accounts with Thunderbird 24.0. 
There is no known limit on the number of address books you can have or the number of addresses in an address book. However, for large numbers of contacts at some point you should consider putting your contacts in an LDAP server instead. (One user has 30,000 addresses spread across 63 address books. . Another has 90,000 addresses in one address book and the address book has become so slow it is unusable. )
Although there might not be limits on the address book size or the notes section in a card, you may easily encounter performance problems when you reach several thousand contacts or have many large notes. The reason is that for every address book that gets used while running Thunderbird, all the contacts in that address book are loaded into memory and stay in memory. This can result in general slowness of Thunderbird, not just slow address autocomplete. (One user reports a 4,265 KB address book of several contacts, including one with a 4.2MB note. It took a long time to switch to that address book or to switch between one of the normal cards and the one with a 4.2MB note. However, it didn't seem to effect auto-completion when composing a message.)
Thunderbird doesn't appear to impose any limit on the number of attachments or the maximum size of the attachment you can send in a message. Any limits are due to your e-mail provider or SMTP server. Note that sending a binary file as a attachment increases its size by a third due to base64 encoding sending two 8-bit characters as three 7-bit ASCII characters.
Thunderbird 1.5 is incompatible with Outlook in how it stores the filenames of attachments when they exceed 64 bytes. This article provides more information and a workaround.
Thunderbird disables cookies by default, but they can be useful for RSS and the Webmail and ThunderBrowse extensions. You can copy and reuse the cookies.txt file from your Firefox profile. Firefox defaults to a limit of 50 cookies per site and 1000 total, which you can change using network.cookie.maxPerHost and network.cookie.maxNumber . Since Thunderbird doesn't support those preferences the limit appears to be the default value in Firefox.
There appears to be a limit of approximately 60 addresses when sending messages if you enter each address separately. However, if you use lists (mail distribution lists) for most of the addresses any limit is imposed by whatever SMTP server you use.
You can receive messages sent to multiple addresses in the same mailbox. Any limitations would be due to your e-mail provider.
A file descriptor is a data structure used to access a file, normally only used with POSIX operating systems. Supposedly there is a hard coded limit in Thunderbird of 256 file descriptors under Solaris. If you open 256 different folders you'll run into that limit, even if you closed most of the folders. Nobody has mentioned a similar problem on other operating systems.
Folders and messages
There is no known limit on the number of folders or the number of messages. However, there are folder space/size limitations as noted below.
Messages are stored as mbox files in a profile. Mbox combines multiple messages of a folder into one large file. A folder needs to be periodically compacted to remove any deleted messages. Compact reduces the size of the folder file as a side effect.
The Thunderbird profile is by default on your boot disk (or perhaps somewhere else if you have moved the location of your OS's "user data"). You are of course limited by the amount of free space available on the disk where your mail is being stored. However, you can use the Profile Manager to create or move a Thunderbird profile wherever you want, including a file share on a different computer. It is also possible to store mail folders outside of the Thunderbird profile, on any drive, in any directory (with some restrictions which as of 2013 are enforced by Thunderbird).
Prior to version 3.1 the maximum folder size is 4GB, or as limited by your file system. This is the same as the maximum file size under FAT32. NTFS (Windows) , ext3, ext4 and ReiserFS (Linux) , and HFS Plus (OS X) support much larger file sizes, and a "All Mail" folder in a Gmail IMAP account is supposed to contain a copy of all messages in the mailbox (which could exceed 10GB) so eventually the limits were changed. Thunderbird 3.1 added a a 64-bit offset to the .msf file allowing imap folders larger than 4GB, if the file system permitted large files  . This change did not help non-imap folders, for example those in Local Folders, including any offline storage (Tools -> Account Settings -> Synchronization & Storage ->Message Synchronizing) for remote folders. Offline storage uses mbox files, just like POP accounts and the Local Folders pseudo account.
tb-planning mailing list posting proposes to "just admit that local mbox folders are limited in size to 4GB, make sure that any usage of mbox folders does not allow >4GB operations, and put our efforts instead to finishing the remaining issues in maildir support."
(Not finished/Not recommended for use) Thunderbird 12 added a pluggable storage API which the maildir storage format (and the current Berkeley mbox). Maildir stores each message as an individual files, i.e. does not combine messages into a single file, and thus bypasses the 4GB folder size as well as eliminates any need to compact a folder. The maildir support, also called maildir-lite in Thunderbird, is very similar to, but not meant to be 100% compatible with, the maildir spec. It was hoped that maildir would be enabled by default in Thunderbird 24.0 (Fall 2013) but it is still too buggy for normal use.  . One item needed is a solution for 32 bit nsMsgKey which is "used in way too many places as a way to access the particular message in a folder". The architect has stated "It should not be used as the offset into the Berkeley mailbox file. If it were [used, then] maildir support would not work at all, instead of only kinda working." Other 4GB folder problems that need to be fixed are making sure that a inbox folder full message occurs rather than losing data and the ability to compact any folder. Maildir also needs a rebuild index mechanism since power uses may fiddle with the contents of the maildir folder from the command line or a file manager.
The support for folders larger than 4GB appears to be broken in recent versions (such as 16.0.2). . There has been a lot of confusion over how wide spread the problem is, and there are no tests for it in the Moztrap test case management software (it replaced Litmus, which didn't test it either). There has also been confusion due to a bug (now fixed) that caused repeated downloading of messages for a offline folder in a IMAP account due to changes in how Thunderbird determined the real message size. 
There are old bug reports for some versions before 12.0 (mainly for OSX and some BSD systems) were the maximum folder size was accidentally lowered to 2GB. There were also some scenarios where Thunderbird didn't check that adding a message would exceed the maximum folder size before it added it. Normally whether or not a folder is corrupted has no effect on the maximum size of the mbox file. However, in a very few cases the corruption caused Thunderbird to ignore the maximum folder size, once reaching 35GB .
Compacting a folder larger than 4GB is extremely slow.
Desktop search programs such as Google Desktop , Copernic or X1 may not be able to index the contents of a 4GB folder. X1 supposedly runs into problems at 1GB.
The depth of the folder hierarchy is limited by the path to last folders header. That can't exceed the operating systems maximum path length (usually at least 255 characters). That is not the same as what you see in the folder pane, its includes the path to the parent folder in the profile (which might be over 100 characters), and five additional characters (back slash plus .sbd) for each child folder. If you run into this limit you can display more folders by moving the profile or the accounts directory closer to the root of the drive, so that less of the total path length is wasted. There is additional overhead if any of the folders have non-ASCII characters. For example, if its a IMAP account the server would send the string as modified UTF-7, which more than doubles the number of characters.
Thunderbird still has a very old bug (from 2004) that prevents it from displaying a child folder if the total length of the child folder's relative path exceeds 128 bytes. The relative path is the distance from the top (root) of the profile directory (not the total path length).
With local folders (POP accounts) you normally can't create a folder hierarchy that you can't fully use. However, that's not true for IMAP accounts since you could create it using webmail, and not be able to download all of the headers due to it exceeding the maximum path length. This limit doesn't apply to a Subscribe window in a IMAP account since it just lists what folders exist on the mail server (its not downloading headers).
If you use View -> Sort by -> Threaded and select more than 100 messages it will not list more than 100 conversations in the message pane. If you reach that limit you should see something like These messages take up: 14.4 MB. (Note: 287 messages are selected, the first 100 are shown) if you scroll to the bottom of the message pane.
Message filters and custom headers
The Mozilla message filter specification states you can create any number of custom headers and message filters. It doesn't state how many rules you can have. However, one of the source files has several error messages about a limit of 50 custom headers which one user has run into.
Number of recent folders
The list of recent folders when moving a message via Message -> Move To -> Recent is hard coded to a maximum of 15. The same limit is used for Message -> Copy To -> Recent. 
Open .msf files
Thunderbird loads the contents of the corresponding *.msf file into memory whenever it opens a folder. Starting with Thunderbird 15 there are default settings (mail.db.idle_limit and mail.db.max_open) to limit how many of those files it will keep open, and for how long. This can have a big effect on memory usage if you have tens of thousands of messages in a folder. Its not clear if there is any limit on how many of these files can be open at the same time. One user had 226 *.msf files open. 
If you get a error message like "Unable to load address book file abook.mab. It may be read-only, or locked by another application. Please try again later." and you are running a 32-bit version of Windows you might have run into a limit on the number of open file handles that the operating system supports, caused by having lots of mail folders. Thunderbird opens the *.msf file for every mail folder.  
Quick search (Ctrl+Shift+F) and the Quick Filter bar don't appear to have any limits. Any limits for "Search the web" (OpenSearch) would be imposed by the search provider.
There is no known limit on the number of SMTP servers you can have. Unlike accounts, Thunderbird doesn't appear to test whether the additional SMTP server would be a duplicate.
Windows and tabs
The limits on the number of windows and tabs are not clear. There are what appear to be limits of eight compose message windows, ten content tabs, and ten web search tabs. However, you can create several dozen compose message windows. The limit of eight compose message windows occurs if you select several messages and then press Ctrl+R to reply to them.   
Limitations of Microsoft Windows
In some cases, Windows users may run into a hard limit placed on the maximum amount of characters any path can be. Symptoms may include, but aren't limited to, disappearing mail folders. The limit is usually a length of 260 characters unless an application supports UNICODE paths. Thunderbird doesn't appear to support UNICODE paths.