Bluetooth doesn't even turn on
So, do you mean that when you go to Settings and tap the switch to turn on Bluetooth, the switch goes to the on position for a split second, then springs back to the off position? If so, I've had this happen, too.
I think, but can't verify, that it had something to do with a 3rd party widget (Power Toggles), that lets you place assorted controls like Bluetooth and WiFi on your home screen and notification bar. When I tapped the widget to turn off Bluetooth, then attempted to turn it on again using the normal Nexus 7 settings method, the switch wouldn't stay in the on position. I tried using the widget, and still no luck. I finally rebooted my N7 and everything was back to normal.
Again, I can't verify that it had to do with the widget, and I don't feel like trying to recreate the problem! But I can verify that I've been in a situation where the Bluetooth switch wouldn't stay in the on position.