James,
I'm going to disagree with the folks above. It seems if it were a taillight ground issue, that it would only effect the light with the problem...not both. I think it is something else, but I don't have a clue what that would be.
At first I thought 'mashed wires', where the brake light wire and taillight wire were bare and touching each other. If this were the case, when you applied the brakes, the brake AND tail lamps would illuminate! I'll wait for the smart guys to show up on this one!
Stuart