94 ford probe wont start unless is it being jumped by another car, but the battery is not dead?

i have a 94 for probe and i took it to have the battery and alternator checked and they are both fine, but when you get in the car all the lights are on and the door dings, but when i turn the key everything goes dim and the engine will not turn over, it sounds like it is starting to turn over, but dosnt. what could the problem be.