From LPZ
"2) Delete those mail-server-related items from your login keychain. Restart Mail and enter them anew when requested; check the Remember in keychain box too."
This is what worked for me.
I'm _guessing_ that there is some "crossed connection" between the Keychain application and Mail.app that is causing the problem. I'll guess again and speculate that the problem is that for some reason, certain normally-hidden bits of data (quite possibly "port-related" or "timeout-related") aren't being properly stored within a user's keychain.
This can become quite a problem when updating to Snow Leopard from Leopard.
The solution is to quit Mail and open Keychain Access, enter the administrative password and then open your keychain (to the point where you can delete password "keys").
Then, delete ALL the mail-related passwords - ALL of them. These will have the letters "POP" in them for a "pop" mail account. If you're using an IMAP mail server, not sure what that will say. But it's relatively easy to ascertain which saved "keys" relate to your mail.
Again, delete ALL of them from the keychain. You don't want any "traces left behind" that could foul things up.
Now, quit Keychain Access and reopen Mail.app.
First, test your capability to send mail. I would suggest you compose an email _to yourself_. You will be asked for a password (of course, since you deleted all your stored Mail.app passwords). Enter it, and (as LPZ said) click the "remember" checkbox for the keychain.
If the mail gets sent ok, now check for incoming mail (this is why you sent email to yourself). Again, you will be asked for a password, and enter it and make sure it's saved in the keychain.
Hopefully, this will get things going again.