I don't know if that's possible. I also can't rule it out. I'm having a hard time imagining a scenario where you have enough current to push the plunger hard enough to make a clicking sound, yet somehow not hard enough to make contact sufficient to pass enough current for the starter to spin...
The circuit feeding the starter solenoid has to go through all of these steps:
That's quite a bit of wire, many terminal connections, and the Neutral Safety Switch. Mild corrosion over the years can lead to resistance that prevents sufficient current. De-Oxit and silicone dielectric grease...