No, it is not possible not to receive DNA from all the great grandparents because our parents have their DNAs.
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DNA analysis of blood samples can help determine your heritage, but comparison to a particular tribe or family would be necessary for you to lay any claims of parentage. Typically the US government will only recognize your lineage if you can prove that one of your great grandparents were 50% Native to a particular tribe or tribal group. Each tribe has their own determinations.
If you are speaking of your American Indian roots, to trace the origins it might be possible by using DNA. It would even be more help if you have relatives and they would give their DNA as well.
it's not.You could be Indian or part Indian but not native American.
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DNA has largely replaced finger-prints for indentifying people. Or at least confirmed what finger-prints have led authorities to. Service people in the military today, all must now submit their DNA to the medical corps, so that in the event they become a casualty, they can be identified. DNA science has probably eliminated future "Unknown Soldiers." For detailed definitions of DNA, see site: DNA