In college geology class over a decade ago, the cut-off was 2,000 ft. Anything with less than 2,000 feet high is not a mountain. Granted that most of the Appalachians are under 2,000 ft tall, but there are also a lot over 4500 ft tall too (Mt. Mitchel is over 6,000 ft, and Black Mountain is over 5,500 ft, etc.).