WHAT could this be?!?

I was thinking that whatever is killing the engine is also cutting power to the starter circuit???

Treblig

Off the top of my head I am thinking the charging system, if the alternator is not charging, the car will drain the battery until it stalls, and then it wont crank. This is physically possible. A good battery will build up a charge overnight, then the car may crank and start the next day. This is a scenario that describes what you are thinking. i guess we need more input from the OP.