I'm currently using Dovecot 2.2.33.2, although I've had this particular issue happen periodically over the last few years with other Dovecot versions. It seems to only affect accounts I'm receiving with POP3, not IMAP ones.
What happens is the "top-most" (or, oldest) message in the "cur" folder seems to get "stuck". So I'll, say, come back in after a break to check my email (Thunderbird in my case, latest version), and in one of my folders I'll have 30 copies of the same email (all with the same timestamp), and no newer messages delivered since the timestamp of that message.
If I go look in the "cur" folder, it has lots of messages in it. The "new" folder is empty. Every time I click "Get Messages" in the mail client, I get another new copy of the same email and no other mail. If I go manually into the cur folder and move the top-most (again, meaning "oldest") message out (which of course corresponds to the message that I'm receiving copies of over-and-over repeatedly) then when I click "Get Messages" all of the other messages come pouring into my mail client just fine and everything becomes "un-stuck".
I've done some experiments, and if I copy that same "bad" message back in to cur with its original filename, it gets stuck again. It is an encrypted message, so I tried decrypting it and putting it back in cur with the original name, and it gets stuck in the same fashion even when it is decrypted. I tried putting the bad message into an account that receives with IMAP, just to see if it would maybe keep messages from delivering or something, but new messages come into the IMAP account just fine, even with the bad message present, so the issue seems specific to POP3.
Any ideas what might cause something like that? I've had it happen twice over the last two days (which is unusually frequent, hence my message here), and the message sources are varied. Today it was a normal email message from a person, and yesterday it was a Twitch notification. I don't see anything unusual in the logs (and I did turn on debugging to see if it gave some additional info), just normal POP3 logins and connections closings when I'm clicking "Get Messages".
-David