What's actually happening
Your softener regenerates on a schedule. The timer motor turns a cam, the cam shifts a valve piston, and that piston routes water through the resin and brine so the system cleans itself.
When the motor won't cycle, that whole chain stalls. Either the motor isn't running, or it's running against something that won't move.
On older units, the second one is the usual story. The motor is fine. The valve or the gears turning it have quit cooperating.
The good news: your house still gets water. The softener just isn't softening it. So you have time to diagnose this right instead of guessing.
Why it usually happens on older units
The timer motor in most residential softeners is tiny. It pulls a few watts and spins slowly. Left to itself it'll run for 20 years.
What wears out is everything it pushes against. The drive gears are plastic, and after a decade of daily-ish cycling they get brittle. One good jam and the teeth shear.
The valve piston is the other weak link. Hard water and brine leave mineral residue that hardens inside the valve body. Over years it turns the piston's smooth slide into a fight.
Put those together and you get the classic failure: the piston stiffens, the motor strains, and the weakest gear gives out. Now the motor hums but the cam sits dead still.
Silent motors do happen, usually after weeks of straining against a jam. But if you catch this early, the motor is almost always still alive.