So I've experimentally started my IMAP client at work after explicitly killing the home IMAP client. What seemed to be happening is that there are two IMAP processes running on the server. One's PID is staying constant and the other is killing itself and restarting.
I tried sending myself e-mail to see what would happen next. The mail simply disappeared. It's not in /var/mail, nor is it in my personal mbox. It's just gone. Also, both of the old imapd processes died, and there's a new one running (connected to the work machine).
At this point, I'm just going to disconnect everything and scan my personal mbox file by hand for the day. Yuck.