When I go to my "mail" I get a box that says "Cannot get mail, check settings....." while at the same time, my mail is delivered behind the box.

Have you tried a reset?

Close all open apps by double-tapping the home button, then swiping up and off the screen with the app window (not the smaller icon).

Reset your device: hold down the home button along with the sleep/wake button until the screen goes black and you see the Apple, then let go. (No data loss)

Have you checked with TWC to make sure you're using the correct settings required to use their email on a mobile device?

Your user name and password are entered into the incoming and outgoing server settings?