Through the cries of some users at Apple's official support forums, we learn that a bug with Apple's Mail client can surface when upgrading to the latest version of OS X, Mavericks. And, though it's a simple bug, it's a frustrating one: Email will not refresh (eg: no new mail) unless the app is closed entirely and then relaunched. Fortunately, Apple's aware of the issue.
While a fix is being prepared, Apple's offering up some advice to help sooth the pain in the interim. When the "Take all accounts offline" option is used, you can use the "Get all new mail" option which will force the client to grab all new email. Apple further recommends creating shortcuts with both options on the toolbar so that it can be accessed much quicker. This remains a tedious solution, but it's a little bit better than having to relaunch the entire app.
It's not entirely clear at this point which email providers are at most risk for this happening, but it does seem to be IMAP at the root of the issue. Some have complained of the issue while using Gmail, while others have experienced it with other providers.
Apple hasn't mentioned when a fix for this issue would be made available, but given how simple the bug sounds, I'd imagine it won't take too long for it to be rolled-out.