Two problems related to wiring have been reported for the 2001 BMW 330. The most recently reported issues are listed below.
I was receiving a intermittent left tail light warning indicator and assumed it was a bulb that was going out. After some time had past I received a steady left tail light indicator and when I was about to replace the bulb I discovered the bulbs was fine and found the wiring/connector in the tail light bulb carrier had a short/melted. I believe this is the same issue/problem on the 2002-2005 BMW e46 3series which a tail light recall was issued.
Left tail lights stopped working. Inspection found burnt ground wire at wiring connector. Replacement of ground wire, connector and bulb holder assembly needed.