How many american players in basketball play for different NT?
max one for each country, and this is mostly based on the limitations of foreign players in lot of european leagues. Thats why lot of player search for european uncles/grand parents etc. to get a european passsport, and that this don't take this long they often had to commit to play for the NT teams(and often don't play a minute in their country for clubs).
Tehri is also the possibility to get it through marriage, but i don't thing that everyone marriage after beiing x year in a country and even then you sometimes have additional restrictions before you could change your nationality.