You have 2 popular possibilites. On the older GE microwaves they had a small circuit board I call the Power/ Relay board. This board is notorious for developing cracks around the solder connections on the back of the board. It will sometimes be intermittant for a while as it makes and breakes connection. Typically it will eventually get so bad it will flash and blow away any solder left on the connection. Repairing consists of disconnecting power to the unit, removing the board (it has the fuse mounted on it) and resoldering and bad connections you may find.
The other very common problem with ANY microwave that only times down but doesn't run at all is with the door switches. For some reason, microwave oven manufacturers like to use switches with small 3/16ths terminals and try to run 12 -13 amps through them. As these connections expand and contract from the amperage causing them to get hot then cool again when the unit shuts off, the connections get loose and get hotter over time as they get looser. Eventually they melt part of the switch and the switch contacts open fooling your microwave in to thinking the door is open, so it will no longer run. The repair of this problem consists of not only replacing the switch, preferably with one with 1/4" terminals, but also replacing the wire terminals on the ends of the wires where they plug onto the switch.
James Bennett / A.C. & M. Service, May 2007