Like others have said, actually check the battery connections. All of them. Don't just pull on them, disconnect them, wire brush the terminals and hardware, and put them back on.
A few weeks ago, my son's friends car wouldn't start. Dome lights and buzzer - ok. Headlights -- ok. Turn the key, everything shuts down. He's been having troubles with it anyway, and was about ready to call a tow truck. I took the seemingly tight side post terminals off, cleaned em up, and put them back on and zoom, she fired right up. No problems since, and the other headaches no shop could fix all went away too.
It is very possible for a seemingly good connection to work sometimes but fail on a large draw.