your right Dennis, I was think of the system on my daughters Malibu ...
It won't start at times with my set of keys, works fine with hers. Same thing used to happen with the Lexus at times, new key cost me $500
Is (or was) it showing the check engine light? Or can you even tell if it doesn't start? I've never seen that.
I have a OBDII scanner if you can't find one locally.
If you think the starter isn't engaging I would start there. I know my 2003 GM will not crank the starter more than a few seconds, then it cuts off and you have to wait a few minutes until you try again - only happened once in the mountains at @-10deg.
If you think it's fuel you can insert key and with the door open, and listening to the rear near the fuel tank, turn the key to the run position (not start) and see if the pump goes through a few seconds of operation to pressurize the fuel rail. Usually you can tell if the pump at least runs this way.
Cracking open the fuel line at the pressure tap fitting is another way to see if fuel pressure is pressent at the rail. (do this with the key 'off' and an "fire putter-outer" nearby ... )
I have also squirted a little fuel into the throttle valve intake (take hose off) to see of just no fuel was keeping it from starting.
I had a 2001 Impala as a lease for a few months, but it always started.
Hopefully someone has had the same symtoms on the same car and will chime-in soon.