I had a similar problem last summer. It was a bad connection in the ground cable. It had a clamp on connector for the battery - one where the cable attached to the connector by a clamp with two bolts. I finally figured out that when the high load of the starter came thru, there was arcing inside the clamp where it apparently wasn't clamped tight enough or something. I took it apart, cut the bad end of the cable off and installed a new connector that I bought at the auto parts store. Solved the problem. The only hard part is cutting a 3/8 inch thick cable! I finally just started pulling several strands out and clipping them, and worked my way thru the whole cable.
The crazy things: That connector had been on there 9 years without a problem, and before I changed it out, I could get it to start up by removing and replacing the battery connections. Anyway, it was a simple fix. And, when I took the old connector apart, the wires inside the clamp were burned pretty badly.