While "native American" could be correctly applied to anyone born in the U.S. (or,perhaps, in the Americas), my understanding is that the term "Native American" (note the capital N) arose from some government project/study that referred to "Native American Peoples". Such peoples include the various Amerindian tribes, Eskimos, Polynesian Hawaiians, etc.--the pre-Columbian inhabitants of the United States' land. It was not simply a PC substitute for (American) "Indian", though it did eliminate the confusion (with Asian Indian) and possible possible negative connotations of that term. While I have no philosophical objection to the phrase "Native American", I find it rather unsatisfactory for three reasons: it's unwieldy; it's indistinguishable from "native American" in speech; and most people are unaware that in its original application it included Hawaiians, etc. I prefer "Amerind", but it never seemed to really catch on.

One American Indian of my acquaintance indicated that the preferred term would indeed be the individual tribal name. This idea, unfortunately, presents its own problems. Most of us can look at an American Indian and indentify him as such, but we don't have a clue which tribe he may represent. What would you call someone whose heritage includes more than one tribe, and perhaps also some European ancestry? And it's still useful to have a term that refers to all American Indians, not just one tribe. As previous responses indicate, there doesn't seem to be a universally or even widely accepted term to use here.