Yep IO has got it in one line:
If the alternator, or its rectifier, or its voltage regulator, fails then you are running from the battery until it goes flat.
Practically that is the reality, and dont forget it. I have had two alternators fail in flight and landed before the battery went flat but I am in no doubt it would have gone flat. I had an alternator fail in my Caterham a little while back and sure enough the battery went flat and the car stopped - of course at least the aircraft engine will keep running unless it is a Thielert.