Recognition in a tribe is not always a black or white issue and there are exceptions to this rule, such as instances where one is raised in the culture or on a reservation but does not have other requirements for membership such as a blood-quantum requirement. This statement is not speaking to those circumstances.
Interesting post, but the info about the DNA test is misleading. Due to the fact that not every child inherits every gene from his parents, etc., it is quite possible for a sibling or a first cousin to have Native American (or any other) DNA markers when another one does not. The fact that you do not have a particular type of DNA does not mean that you do not have an ancestor with that heritage; it simply means that in the gene lottery, you did not get those particular genes from great-grandma, or whoever it was who had that ancestry. That is why old fashioned genealogical research with documents and cluster DNA testing of several siblings or other relatives is more helpful for determining your actual heritage.