The Joshua Duggar scenario is certainly one where it is possible to prove he did it, but consider this (hypothetical) scenario:
The person who used my burner email to sign up pays for his Ashley Madison subscription with a pre-paid Visa or Mastercard, the kind you can buy at any Wal-Mart for cash. My (nonexistent) SO knows about my burner email. She Googles and finds any one of the number of sites that allow you to put an email in and see if it's in the leaked database (without showing supporting info such as the payment info or addresses). Of course, mine turns up, but I'm assuming my theoretical SO is smart enough to realize that email alone is not enough to prove anything.
The SO then looks at the full database, and sees that the account was paid for with a pre-paid card with no billing address. At this point, I have no way to prove that I didn't buy that pre-paid card short of opening up my financials and accounting for every single dollar to prove that there's no way I could've bought that pre-paid card. The possibility that I cheated now exists, and the possibility alone can be enough to cause a breakup or divorce for something that I never did in the first place.
Sure, the scenario is a little far-fetched, but it's not outright impossible.