You know, I just thought a bit about this.
If you are an African-American, you're not an American.
If you're an Irish-American, you're not an American.
If you're an Asian American, German-American, Italian-American, Native American, Arab-American, Jewish-American, Hispanic American, etc., ad nauseum, you're not an American.
To be an American is to place yourself primarily as a member of our group, not to separate yourself with other designations or labels.