There are actually quite a few people in the United States
who are ethnically Asian
, but who insist on calling themselves "Asian-American
" and hate being called "Asian."
The distinction might sound silly at first, but the reasoning behind it is that the term "Asian" implies that an individual is from Asia or living in Asia. Because many people of Asian descent have lived in America all their lives, they don't like having a designation that makes them sound like immigrants or tourists.
Furthermore, some self-described "Asian Americans" don't identify with more recent arrivals from Asia. Many actually resent Asians from Asia, because they feel that other Americans confuse the two groups too often: expecting that people with an Asian appearance will know minutiae about Asian culture or speak an Asian language (or not speak English), for instance.
This dichotomy is only prevalent in the East and Southeast Asian community: Indians, Pakistanis, Iranians, Arabs, and other South and West Asian groups don't have the same issues, largely because they don't have as long of a history in the United States as East and Southeast Asians do.
The Asian/Asian American question can often reach moronic levels. Here at the University of Florida, for instance, a large coalition of students from various Asian backgrounds have been working on a center for Asian ethnic and cultural issues. When the proposal was being drafted, different factions clashed over whether to call the end product "Asian" or "Asian American." The compromise, to call it "Asian and Asian American," satisfied nobody, and there have been several shouting matches since the compromise over renaming the center in one direction or the other.
In the end, there's only one solution. As a friend of mine put it, "we just have to keep fucking each other until nobody knows what anybody else is."