1991 Nissan Maxima Starting problem

We have had this Maxima since new, love it, had the engine rebuilt and a new radiator. The problem we can't seem to solve is an intermittant starting problem. The car can go for months and then all of a sudden, you can stop, go into a building for a few minutes, come out and it won't start. If you try a few times, nothing. If you leave it alone for an hour or two, or perhaps overnight, it starts upimmediately then may not a week or even a few months before it doesn't start again. When it won't start it sounds like a higher pitched starter than on a usual start without "turning over. Any help?