I've run into this same problem, as has a friend of mine, since we updated our iPhones to OS7. However, I don't recall whether either of us dropped our device or subjected it to shock.
As I recall, I was able to resolve it by powering down and rebooting, but it sounds like you tried that already. You might also try reseating the SIM chip if you've got a model with a cellular connection, and you should probably do so when it's powered down. (I'm not certain whether that matters.) You could also try hooking it up to your computer and doing a reset-and-restore-style operation.