Well, that is very close to the meaning of 'nation', which is why international relations students often talk of the nation-state; the nation is the cultural community, the state is the national-level government.

Some nations have no state, such as the Kurds.

Some states have multiple nations within, such as the USA (which is where the hyphenated-American thing had actually started; it made perfect sense to speak of a German-American or an Irish-American, because these persons would often merge elements of both nationalities).