I get this problem with Thunderbird 126.96.36.199 on Ubuntu Jaunty (launchpad bug #444732)
I was downloading many (I think about 8-10) attachments from a message in Thunderbird manually by right-clicking, save as, because the save all button was broken (LP Bug #303140).
I was going through it rather quickly - right click, save, right click, save etc, and expected a queue of message downloads to start (or perhaps them all to come at once).
What actually happened was several partial downloads. From what I can guess, as soon as I told it where to save the next attachment, the previous one was interrupted, and left unfinished.
This was on an IMAPS inbox to googlemail servers.