It is very culturally based.. In one generation, you become a hyphenated american (irish-american, german-american, chinese-american.. some people even drop the hyphenated part sooner.. in our law--if you are born here, you qualify for everything-- including growing up and being president.. I don't think we have had any first generation american as president.. but we could..

In other cultures.. in 500 years, you still don't reach hyphenated status.. Koreans in Japan, Roma's in most of europe, Jews in many parts of europe.. In most countries, with most immigrants, the time frame is longer than US, but shorter than 500 years..

But i have a cousin (by marriage) i thought to be italian.. he looked italian, he spoke italian, his father came to US from Italy.. but he claims to be spanish.. He came from a family that served as servants to some royal household from spain and move to italy with the spanish court. .. and until spanish civil war, the family was still working as servants in government houses (embassies, etc) in Italy, for spanish government. His father spoke spanish as well as italian. the family lived in italy for over 300 years.. but they still considered themselves "Spanish"
(I no idea how the italians in italy felt about the family-- but italians in US considered them italian.) so sometimes its not the host country.. One complaint about the gypsy's is that they are not interested in giving up their language or ways and don't want to become part of the local population.. (or course, by now, attitude have become hardened.. who knows what it was like 1000, or 500 years ago..)