What this sounds like is you have a "bad" new alternator. I'm guessing you bought the reman one cause it was a few bucks cheaper, keep in mind that the reman one went bad once before. I think if you swap out the alternator with a new one your problem will go away. The only other thing is if you have an under drive pulley on the alternator(if you robbed the alternator off your friends hot rod), this would cause the same issue because it would require a higher rpm to "excite" the alternator and begin charging.
One thing to consider would be the connections to and from your alternator.
You might have some corrosion at the battery terminals or even inside the cables themsleves. At times there may be a good connections and other times you may have a bad connection.