I had something like this happen to me recently. Battery was fine, starter replaced, and still acting up. Sometimes it would start, sometimes it wouldn't, sometimes the engine would turn over but never start.
I had a shop add an additional ground strap from the negative terminal of my battery to a bolt on the alternator. Since I have done this, it has started every time. Hook up a jumper cable from your negative battery terminal to someone elses' negative battery terminal, or a ground on your frame. if it starts up with this cable attached, then your battery needs to be grounded better.