Are you sure they're not rabbits, ffs? That's just not right!
They're Amish... up to you to decide the degree of sexual differentiation between them and rabbits
As the story goes (and I have no reason not to believe her -- her grandfather was Amish but left the church and was shunned, even at his funeral) the only reason the cousin hasn't had more is because at various points she physically banned her husband from her bedroom for a year.
Amish men get married early, their wives have kids until they die (generally during or as a side effect of childbirth), remarry, and repeat. It's a very pre-Industrial revolution mindset, since you need a lot of hands to work the farm (and they still do, although as I understand it they work a very small plot of land nowadays and sell the rest out to modern farmers), and child mortality was high. Thing is, neither child nor mother mortality is that high now -- many Amish will still accept modern medical care (if you can get them to a hospital in time that is), and we can prevent most of the deaths that would have occurred otherwise.
So now, instead of having 10 kids before your wife dies, and only having 5-6 of them survive to adulthood, everyone lives. Except, perhaps, the poor mother who has to take care of all the rugrats (er... woodrats? No carpeted floors after all...).
Yes, I now know far more about modern Amish than I ever thought I would thanks to my coworker.