Back-up light bulbs DO fail, but it is not likely that both would fail at the same time.
I have a 92 325i E36 with a MANUAL transmission and 159,000 miles on it and have had to replace the back-up light switch twice. They are cheap ($11) and not too hard to put in. They are located on the side of the manual transmission (I am unfamiliar how the equivalent function works in a car with an automatic transmission.) Working underneath the car, you just unplug the wiring harness, unscrew the old switch, screw in the new switch and re-connect the wiring plug. As I recall quarters are tight around the switch.