I've had that problem, although not recently. In my case it was - as you suspected in your case - a dud message that had given the server hiccups. I ended up reading the mail via webmail on the server one by one, deleting the ones I didn't need to keep and forwarding the others to a different e-mail address. [Edit: gmail ones are free, IMAP, and work well]
Changing to IMAP may do it. If you go to the offending mailbox in Thunderbird and click on the name of the account in the left hand box - the line above "Inbox" - you should see a pop-up showing Email / Accounts/ Advanced Features. Click on "View Settings for this account" and then in the left hand box on "Server Settings" for the relevant account.
You should see "POP Mail Server" or "IMAP Mail Server". POP is Port 110, IMAP is Port 993. If yours is already IMAP, that will tell you that there's nothing to change (POP servers won't accept an IMAP connection).
Unfortunately, changing from POP to IMAP isn't a case of changing the Port Number. There's a Mozilla instruction
here.