As TFSmith121 states, it is almost impossible for a fully sovereign and culturally distinct Native American nation to survive after 1815.
Speaking of North America, I think the most likely opportunity might involve a combination of (1) native tribes adopting the structural, governmental, and technological attributes of Anglo-American culture (such as the Cherokee or others of the "5 Civilized Tribes), that (2) somehow manage to become a more valuable as an independent buffer between two competing European (or Anglo-American) settler societies than as land for direct settlement. This second requirement is tough, given the general and fairly quick rapprochement between the new USA and Britain, the fact that the US was an extremely aggressive settler society, and that the USA soon obtained a clearly dominant position vis a vis other powers (Spain/ Mexico, France) in North America.
However, even with the best conditions, the end result will not be a fully sovereign and Native nation. Given the nature of 19th Century imperialism, a sovereign Native American nation would probably soon
-(like Hawaii) find its native population outnumbered by Europeans and others allowed to move in to establish businesses and provide modern and labor services,
-(like the Cherokee) find its native power elites diluted by intermarriage with whites, and linked to foreign economic and political systems,
-(like almost everywhere) find itself eventually facing a situation where it is absorbed into the USA or another non-native American nation.