1991 Toyota Camry- intermittent ignition problem

I have a 1991 Toyota Camry that run great with approx. 160K miles. Within the last six months, I've experienced a severe intermitent ignition problem. I turn the key, and all the dash lights work, headlights work, horn blows, everything works, except there is dead silence as far as the ignition is concerned. No cranking, no ticking, no noise whatsoever. However, if I keep trying,it eventually will start. My mechanic suggested that it could be the starter, so we replaced that a month ago. The problem continues, however. Now he has suggested that it could be the 'neutral switch', and has suggested we replce that. I have noticed if I play with the shift lever a few times, it seems to start-up, however, I'm not sure that has anything to do with it. A further hint, my back-up (or reverse) lights don't work, however, my mechanic doesn't think it has anything to do with the problem. Any ideas or suggestions would be deeply appreciated. Thanks. Ronnie