Recently I saw an article in my language (Serbo-Croation) that has caused quite a stir for some people. Basically this Serbian guy who thought he was a Serb all his life took a DNA test and found out his DNA is pretty much Bosnian/Montenegrin and less Serbian. Throughout Yugoslavian history people have changed their identities over and over. During the occupation of Montenegro lots started to identify as Serbs, and of course that continued to echo on throughout generations, until they take a DNA test and receive the shock. So my questions is pretty much the title, but I understand it might be complicated. Like for Mexico a lot might say that the people who's ancestors came from Spain are Spanish, but they might be more Native Aztec than Spanish due to the mixing between the two.
In short no. Anthropologist here. A big issue in the USA is DNA testing for American Indians. There are three main types of testing; haplogroup, y chromosome and mitochondrial DNA. Haplogroups currently are not well defined and have a small sample size so they're not to accurate beyond really pre history archaeological work. Y chromosome is only inherited from the father if you're a male and mitochondrial is only inherited from the mother. If someone was half Souix, these tests actually have a good chance of saying they're not native. If the mother is European and the father Souix and a mitochondrial DNA test is done the test would read European. But if a y chromosome test is done in the same scenario, only the sons could claim ancestry. Finally if a haplogroup test is done (D is the most common for American Indians) it narrows it down to ancestry from the Americas, most of Central and Northern Asia and parts of Eastern Europe. While these tests can help archaeologists test migration hypothesises they're not exactly accurate for the purpose you've described.
While you might find an answer here, you should consider posting at /r/AskAnthropology as well!