Battery-alternator problem.
 

It's been really cold here (~10 F) over night and my car has developed
a strange problem. It's a 1991 Accord EX StatWag. The battery is 37
months old. It's my wife's car so I only discovered the problem when
I used it the other day. The car started fine but the battery light
stayed on and eventually the car almost didn't start after one stop.
When I got home I measured the battery voltage and it was about 12 V.
I charged the battery and things started fine and light did NOT come
on even as I drove it on a short trip. I measured the voltage at the
battery with the engine running and it's 14.6V. Next morning a
problem again with the charge light. The battery drops down to 12.5V
overnight after being charged. When the car is running the voltage at
the battery is only 12.1V under this low voltage condition. If the
car runs long enough the charge light sometimes goes out and the
voltage from the alternator goes up to 14.6V. So my question is can a
low (going bad) battery make alternator voltage drop, and the charging
light stay on?

Thanks,
Bob