What would make a car start then stop

Just about anything. How long since a tune-up? Any applicable mil codes? You should be able to check those, yourself.

When the problem occurs, if it cranks good but won't start, have a helper crank it while you check for spark at the plugs. If you have spark everywhere, use a gage and check fuel pressure.

As far as overheating, does the fan work ok? Is there coolant flow through the radiator? You can remove the thermostat, just for testing, and hook everything backup to check for coolant flow. You don't have to wait for coolant to get hot.