Starters have been known to develop dead spots in the windings as they wear. If the batteries check out okay and there are no obvious issues with the cables or terminals, you might try smacking the starter with a hammer to see if it wakes up.
If that doesn't correct the problem, you may be looking at a bad ignition switch, bad solenoid, or possibly some sort of safety interlock. Check connections at the starter, the ignition switch, solenoid, anything to do with the circuit. Also check to make sure that the start circuit does not have a secondary control relay inline to the solenoid or something.
Most starters can be jumped out to pinpoint a fault in the starter versus a fault in the control circuit.