I had a similar problem a few years ago. I had somehow installed a taillight bulb (#1157) into the backup light socket which gets the #1156 bulb. The bulbs and sockets are mistake-proofed so technically it should not be possible, but I did it. Double check that your taillight bulbs are #1157 and your backup light bulb is #1156. If that's not the problem, then I'm with the others regarding the grounding.