I had an intermittent start problem that I think resulted in me replacing an alternator when it was not needed. The battery cable from my solenoid to my positive post on my battery had a "do it yourself" battery post terminal bolt-crimped to the terminal. The cable had corroded under the bolt-down crimp and while it felt strong it was failing electrically. It allowed enough power through the connection to power lights and the door-open buzzer and key-in-ignition buzzer and such, but when you tried to start the engine, nothing. Even when you ganged in the house batteries, still nothing. This should have been my clue. The problem was not the batteries (which were fully charged), the problem was the connection from the engine battery to the solenoid. So even ganging the batteries could not overcome the poor electrical connection.
I replaced the foot-long cable with a new one with factory-crimped-and-overmolded connections. The battery cable from the negative post was also fraying (and much harder to replace as it went down beside the engine to wherever it connects) and I had my mechanic replace it also.
So I would check your electrical connections. Take them apart and clean them.
Steve