I have a 2004 Ford Mustang GT. When I purchased the car the battery light was on. I replaced the battery and the alternator as well. Test it with a meter to confirm 14 V were coming out of the alternator. Drove the car for a few miles light came back on.

I thought OK maybe it was a bad remanufactured alternator. I replaced it with another one same issue. In addition I confirmed the second alternator was receiving 14 V after it was installed. After a few miles of driving light came back on in the meter read 12 V from the battery to the alternator.

however if I rev the engine past 5000 RPMs the light goes off

 

Most remaned alternators, especially store Brand ones are junk. Duralast comes to mind but any of them fit this category. They typically only replace the part that have gone completely bad and not ones that are borderline or ones that wear out (stator) unless they caused the failure.

Go with Bosch remaned or a new aftermarket one like PA Performance. Pricey but worth the value in terms of reliability.