Thunderbird downloads messages on one machine, but not the other.

My desktop stopped downloading messages, but my laptop does fine. The desktop states that the server responds with "no new messages", and it stopped working about the time I set up my laptop to download them. When I go to the web-based interface, the new messages are still there. Running in safe mode doesn't change anything. Some details: both machines are Ubuntu, they started with the same profile, and encrypt the database. Desktop: Ubuntu 12.04, Thunderbird 31.1.2. Laptop: Ubuntu 14.04, whatever is latest and compatible---I don't have it here. I suspect that it is some setting telling my desktop that these were already downloaded because of the laptop, but I cannot find that setting.