I doubt it has much to do with "run in background".
It's possible that there's been a change to server name and/or port number, these things happen.
Please long press the account (on AquaMail's main screen) -> account setup -> Manual -> Next
You'll see detailed server settings like server name, encryption type, and port #.
Feel free to post them here (without the login or password, of course) and I'll check if I'm able to connect to this server (just connect, will not try to log in, obviously).
Or you can try to find server information (server settings) on your mail provider's web site and see if they're changed.