1999 Ford F-150 Truck won't start

My husbands truck will not start after being ran and getting hot or being in very hot weather. We have replaced the starter and gone to the dealer to have a $100 diagnostic done and they could not tell us what was wrong. It was not acting up when they tested it. It had been sitting for about 4 hours before they did test it, so it was not hot. The dome light, dash board, and head lights will turn on, and if the door is open there will be a dinging noise from the key being in the ignition; but when you turn the ignition key, there is not even a sound that happens at all. Just nothing, almost like the battery was disconnected but, well, with everyting else working, it is not the battery. And it is not the starter, it is new and when we took the core back, they tested it and said that it was fine.