well it seems to be a common problem it might just be the switch on the transmission
Check the backup light switch.
check the ground for the lights
it is a 2001 Chevy impala
Bad switch.
It could be the dimmer switch that will adjust the dash lights
The backup lights are a different bulb, then the turn signals and brake lights and The Number 1 reason would be that the bulbs are burned out and need replaced. Good Luck and Remember.Change Your Mind, Not Your Oil.Use the First In Synthetic Motor Oil's.See My Bio For Information & Contact Details.
check fuses. check bulbs. check wires. If not any of those, good luck.
No brake lights but turn signals work? Turn signal switch brakes route through it.
Check the fuses Check the backup switch Check the wiring - make sure that there is good contact.
on some cars the tail lights and brake lights are totally different bulbs gain access to the bulbs and have someone push the brakes so that you can determine which bulbs are the brake lights and replace the tail lights
Be sure the dash light dimmer switch is not turned off and the fuse is okay and the bulbs are good.
check wire continuity- no power no lights