On my seldom-driven car, the first twist of the key after a week or two of sitting is usually met with a "click" as if the battery was only strong enough to actuate the solenoid and not spin the starter. Or maybe the ring and pinion were contacting but not engaging (I doubt this, but it's not impossible).
So I took the starter out (what a hassle!) and bench tested it; it worked fine. What I now suspect that the solenoid, which (1) pushes the pinion gear into the ring gear, and (2) switches current to the starter motor, isn't working correctly i.e., (2) isn't reliably operating.
The solenoid wires are soldered to the starter, thus to separate the two one will have to melt the solder and pull the wires off (the starter). Too much for me at the time; I greased the visible moving parts and reinstalled. Usually the starter will engage and spin the engine on the second or third try. The rest of the day it works fine and I won't mess with it further. Owners who are more particular about their cars may proceed further.