Rally - Ford is prob. right - have seen batteries with "dead" or "shorted" cells act this way, so you can't assume an alt. prob. (even with symptoms that point that way) without rechecking with a known good battery first.
REALLY bad battery, won't continue running even with a good jumper battery attached.
Sometimes shows as an intermittent B4 it dies completely.
That sounds like a good dealer - they could have sold you on a new alt. without a quibble given the symptoms - just did the cheap fix it really needed...