Multiple accounts on same domain in Apple Mail blocks connection to the mail server.

I am trying to run two separate mail accounts on my Mac in Apple Mail, both using the same domain name. I don't want to use aliases because one is a personal mail account and the other is a helpdesk one, with different signatures. I also want to keep emails for those accounts separate.

I can get the first account working fine. When I add the second account, Apple Mail tells me it can't connect to the mail server for either account. As soon as I remove the second account, the first account starts working again.

Anyone know why this might be?

The mail server is Postfix running on a Linux server. My Mac is running OS X Lion v10.7.5.

The mail accounts are IMAP - I also access them from my iPhone 3GS (iOS 6). Again, I can get the first one working from the iPhone, but as soon as I add the second one it falls over.