I suppose its a bit hard to prove when its possible to be related to these non-related groups via NPE. I've got a few groups in my family that bewilder me but I've always figured that I just haven't found the connection.
My take is there are very few false positives on Ancestry, at least at >= 30cM. I use the search button to quickly look up dead ends in matches' trees, which allows you to patch most of them. Others might need a little digging, and some can't be done without ordering records. If you ignore the ones with no tree or are indeterminable, then I have clusters where I've found the common link. So my take is most if not all above 30cM are real relatives. Not sure about the lower ones.
When I contacted many on Living DNA I found many said they had a tree on Ancestry, where we didn't match at all in many cases. Living DNA matches are at least half false positives in my estimation. Ancestry do a very good job of stripping them out.
As for NPEs, in the medieval time about 1% of children were born out of wedlock, then in the 1500s it went up to about 2-3%, down to 1% in the 1600s, up to 5% in the 1700s and down to about 3-4% in the 1800s. Two studies have put the rate of infidelity at around 1.35%. So in the time period autosomal tests are relevant we are looking at about 5% chance of any birth being an NPE.
We can use DNA to prove our lines are correct, that leaves the other party. At four cousins that is five births with around a 25% chance of an NPE if we take the numbers above. NPEs seem to cluster, due to the morals (or lack thereof) of the family. But I think it's fare to expect around 25% of a ~4th cousin cluster to be coming from an NPE.
If it's illegitimacy, it's more obvious. Using overlapping locations, triangulation of matches and Ancestry's clues spotting a match by illegitimacy is not too difficult.
Infidelity can be a bit more tricky, as you don't have the gaping hole births out of wedlock tend to leave in a tree. But I have found two.
1) I found a match with a birth out of wedlock with the first name Harry Roberts. The test subject was on the same generational level and my great-grandfather and overlapped with one of my clusters where there were only two children who survived adulthood over two generations - one named Harry Roberts.
2) My best friend's father took the ancestry test. He shared the matches and the first thing I noticed was the first surname Ancestry suggested from his shared matches was an unusual one from which I descend. I being sorting his matches and soon come across 56cM match who I knew was also a descendant. As I continue I find more and more familiar matches all from this line of my ancestry. His father did not match my immediate families' tests, but in total there were over 200 matches in the cluster where the common ancestors are also mine. So I was able to pinpoint the shared ancestry from my tree and the rough point where it interjected with my friend's father. I found that at that point there was a Mr. Morgan who married a Miss Roberts. There were DNA matches for the ancestry of Roberts, but zero for Morgan. And Ancestry trees had many with the Morgan family in it, which I use as a measure of whether there will be tested descendants. I found that the first two children born to Mr. Morgan and the former Miss Roberts were actually the children of my 4X-great-grandfather who lived about 2 miles away. The following year he moved 20 miles away and it does no appear any ensuing children were his.
If one digs deeply enough into the matches, and catalogs them thoroughly based on the flow of genes, one can root out NPEs. But the upper limit of being able to do so would probably be births around 1800.
