Apple has offered up an interesting remedy to a persisting bug plaguing the OS X 10.9 Mavericks Mail application which manifests itself by preventing the app from receiving any new messages until it is quit and reopened. Apple's workaround solution is a primitive one, albeit effective.
As Mail needs to reestablish a connection to the troublesome mail server (which typically means Gmail), Apple advises quitting and reopening Mail.
As a result, this forces Mail to re-establish a connection. An alternative solution entails a manual process that reinitializes an affected mailbox without having to restart the program...