Repeatably downloading the same messages
From MozillaZine Knowledge Base
POP3. If Thunderbird keeps downloading the same messages for a POP account the popstate.dat file may be corrupted. Thats a file in the profile folder that keeps track of what messages have been downloaded from that POP3 server.
IMAP. If you are using IMAP, there have been reports that Thunderbird may repeatedly attempt to download the same messages if its copy of the folder has reached the file size limit for your file system. FAT32, for example, has a file size limit of 4GB. If you are storing messages on a file system that has such a limit, and are seeing excessive network traffic when Thunderbird is open, or repeated download attempts of the same messages, then you may want to check the size of the files in the ImapMail subfolder in your profile folder. If the file representing a folder has reached the filesystem limit, some potential solutions in order of complexity and risk are: (a) compact the folder in Thunderbird; (b) move old messages into another folder; or (c) delete the folder's file in your profile folder, and Thunderbird will re-download the messages when it is restarted; (d) move the profile to another disk on your computer that has a filesystem with a larger file limit; or (e) converting the disk with the profile to a more modern filesystem with a larger file size limit.