Did you swap the left and right bulbs to see if the problem with illuminating them changes with the swap or remains the same? If the problem follows the swap, then the bulb or holder is the trouble, if the problem remains the same then the vehicle is the problem.

There are 4 screws holding the tail lamps in the car. On the E36 (3 Series) cars, there is significant trouble with the ground wire on the tail lamps burning up -- physically getting hot and charring. I don't know that this is a problem for the E53 or not, but the inspection is easy. Unplug the wiring harness at the tail lamp and observe the condition of the connector pin where the BRN wire is. If charred, then you found the problem. The cure is to establish a new ground path.