I have experienced the same exact problem you explained. One day, the nav screen was gray and blank. When I turned the ignition on, the screen turned black with a fuzzy white line across it. The radio or cd will play for 2 minutes and cut off. If I push the radio button back on, it will play again for 2 minutes and cut out again.
I had a dealer look at it, and charged $95 to hear the internal nav system failed and a new computer would be $3000. I took my car to another mechanic for a second opinion. It was there that I learned there had been a navigation system failure campaign (84-06-00) on many of the 2000 models. My model was among them. Questioning Customer Relations and then the dealer's assistant manager, I was told yesterday that the BMW records database said it had been modified with the retrofit adapter cable in March, 2001. As I purchased this certified auto from a BMW dealer in December, 2002, I was unaware of the campaign (recall). Now I question if it was truly retrofitted, and if so why did it fail exactly as explained it would in the recall letter if not corrected? Were the early navigation systems "lemons" or is it something else?
BMW Customer Relations wasn't very helpful. He kept repeating that the car was out of warranty and "things" wear out. I've not heard of a navigation and radio system wearing out -- have you?