Outlook sporadically stops checking mail
I have experienced the same issue with Apple Mail since updating to Mavericks.  I was hoping Outlook would be better, but it sounds like it has the same problem.

My understanding is that this occurs occasionally when checking mail hosted via Kerio or Exchange.  My workaround is to use Thunderbird, which has not had the problem.  I actually have Thunderbird running simultaneously just for the alerts, and if there is an alert but Mail hasn't received new mail, I know to quit and restart.