Identical symptoms on my 2000 Chevy Express 2500 van, although only on one side. Upon inspection, the back up lamp and socket were very corroded, and although the lamp was still functional, I decided to replace the tail light harness and lamp for less than $30 at the local auto parts store. This solved the brake light problem as well. I thoroughly appreciate the generous expert contributions on this thread which helped me save a trip to the mechanic.