Do these cars have the anti-theft module with the "special chip" built into the key?
Do you have another key that is newer and in better condition? Try it.
My 95 Lumina would periodically refuse to crank over. I could crank it over by jumping the starter solenoid like aefd124. It would start and run for a few seconds but then stop. I traced the starting circuit to the anti-theft module...it was not allowing current to the starter solenoid. I looked at my key and saw the "chip" contacts were worn down a lot...so much that they were not making contact inside the ignition switch. I changed over to a newer key and the car started immediately!
I took the old key and built up the contacts with a little solder and now it works fine.
By the way, my Security Light came on at odd times like noctorum. That was because of the worn out key. The computer thought I had the wrong key or no key at all.