+1 on the tests not being inaccurate, they just take some understanding to interpret the test results.
So a potentially good way to look at DNA Ethnicity tests is like hearing someones accent and guessing where they are from. The more you understand about accents increases the chances you will guess right, but of course accents can be very complex, people move around and accents intermix... so you might year a bit of a specific British accent in another region of Britain or even the world mixed with something else.
Think of it like this, you have person A who you want to guess there accent... how do you do that, well you get accent samples from all the regions you can possibly think of that have distinct accents. You want multiple samples of that accent so you can hear variation.
Then you compare person A to all those samples and try to find the mix of accents that match. Maybe a bit of Irish plus a bit of Georgian (USA) mixed in so you can tell the person might have been born in Ireland but moved to Georgia and picked up some of that accent.
This is like DNA testing. It's like most science, it works with likelihoods and statistical comparison to guess the most likely answer. Basically certain segments of DNA *look* like ethnicity A or B.
The other thing to consider is that you are only guaranteed to have about 5 generations of autosomal DNA since it's random, so you might not have any DNA come up in ethnicity from certain branches beyond them.
Since you inherit DNA from both parents you may in fact have more of a certain ethnic DNA than both parents. I for example have a higher percentage of "Irish" DNA in my results than either my mother or father, apparently I inherited a good chunk from both.
The last thing I'll say is that many Americans *think* they have Native American DNA but in fact don't, these stories are heavily prevalent and often proved wrong... and often when not proved wrong they are proven inaccurate. Usually if there is any truth to them the ancestor they thought was full blooded indian or half blood was either not indian at all (and another ancestor might have been) or they were much lesser indian than the stories suggest.
This sort of thing is extremely common. I've also noticed that 23andme is better at picking up small segments of Native American (anything below 2% or so) than Ancestry (though Ancestry might be better at identifying West African)... though if you have anything over 2% or so I'd imagine Ancestry would pick it up.