Battery
Could be loose or corroded connection on the battery cables. Or could be the battery. Kinda like when you have a bad connection on your car, the battery has enough power to run the radio but not start the car.
You could take the battery to an auto parts store and have them test it under load. If the battery is O.K. I would check the cables for corrosion, sometimes the corrosion could be where the wire connects to the terminal, like a bad solder joint.
Lou