I was trying to login with a few different mail clients, and it gave me various errors, except for Squirrelmail (Webmail) which worked fine. Logins from pretty much any machine on my network failed, unless it was through webmail.

The error message I was getting in the logfiles was:

Jun 19 23:28:55 behemoth dovecot: imap-login: Aborted login: rip=192.168.50.100, lip=192.168.50.1

That's it. No matter how much debugging I turned on, that's all it would say.

I eventually discovered the problem was that the server was refusing to even speak plaintext UNLESS the process talking to it was local (like Squirrelmail, for example). So the remote clients got immediately disconnected the very moment dovecot realized they weren't asking for an SSL/TLS connection.

You have two remedies available to you. On the client side, you can switch everyone to use TLS or SSL. This is obviously preferable, since it ensures that passwords are never sent in the clear.

However, if you are only using your imap locally or through encrypted tunnels, it's probably safe to use the server-side fix:

In /etc/dovecot/dovecot.conf:
disable_plaintext_auth=no

Then restart dovecot, and all should be well.