Car won't start except when jump started and battery is good?

My girlfriend has a a Ford Explorer sport trac, she has a bad habit of leaving the lights on and killing the battery. Had to jump start it quite a few times. Well, yesterday she did the same thing so I went and jump started it. She was able to drive home with no problems (20-30min drive). The next day her car wouldn't start, not even try and crank, no clicks, nothing. I had the day off so I took her to work and went to test battery at shop, battery is fine. I put battery back in her car and turned ignition to on position and radio and everything worked. I tried to crank it, nothing again, no clicks, no power on radio or lights. I jump start it, it runs and I let it run for a bit, shut it off, try again, and nothing. Just for the heck of it, I went and got a new cheap battery just to see if the shop may have been wrong, they werent. I don't think it's the alternator as it will run for awhile without dying, or the starter because it starts fine if using jumper cables. No corrosion on terminals. Don't know what else to look for. Any help?

I presume the "cheap" battery you bought had the capacity to start the SUV. Take a close look at the battery cables. Even thought they are not corroded, remove them and clean the inside surfaces of the connector with a wire brush. Plenty of oxidation, while not evident on the battery terminals, it can prevent a good, solid battery terminal to cable contact. Even a new battery won't deliver the current through badly oxidized battery connections. We're used to seeing a powdery, granular pile of corrosion on battery terminals. But oxidation is not something that we think of when a car won't start because it doesn't leave that tell-tail pile of corrosion.

Other than the above, if that doesn't work, I would say you have a very unusual problem that I can't solve.

I had the same thing happen years ago with a Ford Taurus. The starter checked out okay, the battery, and everything else. This was on a 3.0L engine. After five times of taking off the starter and twice having the battery checked I knew it had to be something I just wasn't seeing. I removed the starter and even though it checked out okay off the car the parts place was zapping it with a heavy amperage to check it as well as using my cars battery and another car's amperage to start. Take the starter apart and CLEAN it completely or just replace the starter completely. I used to rebuild starters and when I took the starter from the taurus apart I got at least a good pint of dust and buildup out of the inside of it. Using an air compressor I blew ouit the rest of it and shined the armature and brushes to make good contact. A dirty starter will still work if you jump them or check them on a test bench but will fail trying to use a car's battery by itself.