Date registered: May 2007
Vehicle: 2003 Mercedes Benz E500
Location: San Francisco Bay Area
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Take a voltmeter, or test-lamp, and check if current is getting to the lamp sockets.
If not, check the fuses. If the fuses are okay, chances are that the contacts in the neutral safety [NSS], or "inhibitor" switch are bad; these contacts activate the back up lamps when reverse gear is engaged.
The NSS is on the side of the transmission and can be a read pain to change, but it is a doable DIY job if you can get at it.