You're a perfect example of the vast limitations of these tests, and the reason why they should not be used to qualify or disqualify anyone as belonging to any particular group! It sounds like your great-great-grandpa was accepted as Apache by the Apache community, and that your great-grandma grew up in that culture. There's a reason tribes don't accept DNA results for membership. It's absolutely possible to be a member of a group via adoption, marriage, etc. rather than DNA. In fact, DNA (in the context of these tests) is just looking at patterns common to certain groups of people who inhabited certain areas at one time. It's basically telling you, "Your DNA matches about 17% with what we've discovered people from X region tend to show."
In the 18th and 19th century, many Irish people married French Canadians, and many more Irish orphans were adopted into French Canadian families. They were raised French, and the majority of them forgot their Irish heritage. Depending on the location
and random chance, it's possible that a few generations later you could have French Canadians who were actually mostly of Irish descent - and yet culturally speaking, they were 100% French Canadian!
The tests are a lot of fun, and they can rule out certain things at a broad level (I remember one particular woman who was adopted and raised believing she was part Native American, but her test came back 50% Japanese, and she went on to discover that her birth father was, in fact, Japanese) - they can also provide clues ("Hmm, my test came back 25% Jewish, maybe that rumour about Grandpa So-and-So was right"). But they absolutely cannot define your identity! Reality is far messier than that!
Yes, you do, to the extent that you would be able to draw a conclusion about their ethnicity (which is another issue altogether).