It's that last phrase that's the kicker. No cousins they were aware of. If you go back into the family tree, you'll probably find multiple cousin marriages further up the line, say 6-7 generations back. Most are unknown to the descendants. My great aunt married her fourth cousin, and they did not know this until after she started researching D.A.R. stuff. They are both descended from the same Revolutionary soldier. They never had children. Probably no connection, because the rest of the family tree is generously out-crossed.
Problem is, if you have multiple cousin marriages even if very far back, the effect is exponential, not just simply cumulative. What you have is not just "double cousins" (where a couple are cousins through two different branches of the family tree) but triple and quadruple cousins. If there are only 1500 ancestors (not all of whom successfully reproduced) back in the 1600s, by now you have people who are, say, not just sixth or seventh cousins but five or six times over. In the genealogy trade, this is known as "pedigree collapse". Or, in more humble jargon, "the family tree don't fork."
We've seen this in cats and in dogs, when a popular bloodline starts cropping up all across the pedigree eight or ten generations later. Even if you don't have any in- or linebreeding in the five-generation pedigree, if there's multiple use of a single bloodline the defects will start to crop up.
You are absolutely right. It seems that the pedigree is collapsing pretty hard in recent generations because it seems I hear of health problems if one sort or another in almost every Amish or Mennonite family (if not THAT family, a cousin's family).
What I always tell my kids is, say hi to to the Amish because they are your cousins. Now, my it was my great-great grandmother who was the last Amish, but there have been Mennonites and Dunkard Brethren going back to the 18th century and I have traced our family tree back to the 1500's in Germany through one line. I tell my kids that we are cousins because if we go back that many generations we ARE related, we all have some common ancestor who came to Pennsylvania from Germany and Switzerland and intermarried for many generations. My mother's family eventually settled in Goshen, Indiana, another large Amish enclave but they all come from the original stock that settled in Lancaster and Berks counties in the 18th century.
Genetics is indeed a fascinating subject...
What I meant to add to the last post was that indeed, the Mennos and Amish by definition HAVE to be cousins even 6-7th generation abck, or closer. I often ask the ones that I know are related, "Oh, isn't so-and-so your cousin?" And they say, "Naw." That is because they are 3rd cousins. I think they only count 2nd cousins as being related. A very dangerous mentality. Or they will say "we are not related that we know of", even though they share a VERY common PA Dutch name. (Not names that the "rest" of the people out here have, and these are people in their own denomination!!! So perhaps that is where the trouble lies. They need to see how close is close.