Originally Posted by sasquatch

Most likely the problem is the conact area where the buld fits has been "burned" away due to arcing. BMW uses a poorly designed bulb "busing" system in which the bulbs can move or otherwise vibrate within the holder. It causes the contact points to become separated and they'll arc when a voltage is applied. Over time it will literally eat away the metal. Very common issue which is easily fixed.

Remove the taillight assemblies. Take out the bulbs. Take a close look at the round metal contacts and see if you can find any areas where the metal busing looks like it was melted or pitted. Clean the area and use some good quality solder to fill in the holes/pits. Sand, grind, or file smooth and flat with the metal. Reinsert the bulb and reinstall the tailight. Problem solved.