RE: My radio cut out when the motor cut out. What happened?
The BEC (battery eliminator circuit) in the ESC ( electronic speed controller ) is a voltage regulator which supplies current to the receiver and servos. Most voltage regulators will cut out when over heated due to too much current. Many are rated for 2 or 3 amps. The actual cause of this could be due to to a bad servo or too many servos drawing too much current. It doesn't explain why you still had power. If the receiver dropped out you should have also lost power from your ESC to your motor.
I agree that you should have each device checked out and or try substituting each device until you locate the source of your troubles. Good Luck