You still might get a click but there's not enough voltage being supplied to actually trigger the starter?!
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 only way I can see that happening is if it only contacts one contact, due to poorly seated contacts or the plunger somehow being slightly cocked at an angle.
In my opinion, you're either going to get no click because there's insufficient current to plunge the plunger (excessive resistance in the wiring or a bad battery), or you have messed up contacts preventing the plunger from seating against both.