I had a 1994 Buick Park avenue that would do this occasionally . The Mechanic replaced the crank sensor " and it "solved" the problem for several months. After several months a similar problem came up again. This time it appears to be a short in the Battery Cable connections. I could get out and just peck on the cable and it would start. I am going to replace the battery Cables entirely and see if this "solves" it again.
I would start with the Pickup Coil and the Crank Sensor.
what will cause a transmission in a 1992 buick lasebre to shrudderas and stall engine
The rev limiter would have cut in
I would replace the control module within the distributor ( $ 25.00) It is not hard to do. This module is a microprocessor and as they age they are sensitive to heat. As you drive your car it warms up and shuts off.
check crank sensor
Check the fuses.
How does it not start? is it cranking? Could be many things, battery, starter, etc.
theft system
check igntion coils
Could be the immobiliser, try disconnecting the battery for 30minutes a then connect it back up and try starting it again!!
The local Buick dealership would be a good place to take your automobile. If there isn't a dealer near you I would start calling repair shops for pricing. I like my local Meineke shop for various automobile repairs.
sound like when it heats up it shuts off and then when it cools down it starts back up that would be your engine temperature sensor