2003 Ford Taurus Intermittent Starting Problems

My 2003 Taurus sometimes refuses to start. There’s no real pattern: sometimes it’s after the car has run for a while and other times not. Sometimes the starter clicks, other times the engine lights come on with no sound at all from starter/engine. Thing is, the car always ends up starting if I let it sit long enough (15 minutes–several hours). I’ve read around on the internet and realize this seems to be a problem experienced by many owners. I’ve changed the battery, checked all the connections, cleaned the starter connections and the Mass Air Flow Sensor. If the starter was bad, it seems like it wouldn’t start at all, right?

I should have said–starter and battery test within normal limits.