Mail.app and multiple copies of emails
To flesh out an answer from your comment in case other people encounter this problem...

You need to provide the full e-mail address as the username. If you have two accounts with different domains and the same local part and you only specify the local part as the username, it can cause problems. 


First.Last@gmail.com
First.Last@company.com


It may appear that both accounts are 'working' (as in no connection errors) but really it is just connected to the same account twice causing the double messages as you described.