I'm running Mac OS 10.8.1 and have a (personal) Gmail and a (work) Google mail account setup inside Mail client. Every few days (weekly?) Mail seems to forget my personal gmail password and I have to re-enter it again.
This bug seems minor except that with Google 2 Factor authentication, this involves a trip to Google accounts and setting up another password etc. So this bug is hugely annoying when it manifests itself. I don't see this issue on iOS Mail (iPad as well as iPhone) nor on Outlook 2010.
I tried a solution at http://805creative.com/blog/what-to-do-when-apple-mail-keeps-forgetting-your-passwords.html but the plist in question doesn't exist on OS 10.8.1 ..
[UPDATE:] This can happen on either the IMAP (incoming) or SMTP (outgoing) servers. I just saw it happen on the incoming and then 5 mins later on the outgoing. Thankfully I misspelt the label of the outgoing server ('GMial') to notice it was complaining about a different server!