This sounds exactly like what I have been dealing with ever since I bought my car. One day I measured the voltage at the starter solenoid and noticed that I only had about 6 volts there, which apparently is not enough to activate the starter. This meant the voltage dropped a lot when the circuit went inside the car.
I ended up running my own wire with a relay to the starter to a switch inside the car so I could start my car whenever it gave me a problem. I never really found out what the cause of this was, but a little over a year ago I noticed that my car had just over 13 volts while it was running. I figured this should have been somewhere closer to 14.4 so I ended up adding 4 gauge cables from the alternator to the auxiliary post, and from there to the battery. I also added a cable from the battery to the ground. After doing this I've only had the starting problem maybe 3 times in over a year. Now my voltage with the engine running is also at 14.4 or higher.