TLDR; try pulling the battery out and putting it back in.
EDIT TLDR; My phone started to do the screen flickering thing. I followed the directions to bake the motherboard which worked for a little bit. I ended up buying a new phone.
Recently, the Bluetooth radio on my LG G3 started to turn off instantly. I would push the Bluetooth button on the pull down menu, and the little icon on the top of the screen would flicker on and then flicker off a couple seconds later. I Google searched for some help and there were posts about installing weird apps and custom ROMs, hard restarting, and reformatting.
It turns out none of those things were necessary. Before you go off and try all those things, try turning off your phone, pulling out the battery and then putting it back in. My Bluetooth is as good as new again.