There are attack dog breeds like pit bulls that if they've bitten into someone can't let go anymore; it's impossible to them. The neck-breaking thing is something that happened years ago in the part of the city I lived in; a dog had attacked someone and a woman had run out and used her broom to bend back its head until, well.
It's basically one thing you can do with the kind of dog that is impossible to get to let go.
There were a lot of people owning attack dog breeds in that part of the city. It wasn't an unusual sight to go to an off-licence and have guys standing there bragging about the bite force of their dog to each other.
So, you know, it's not that I generally condone neck-breaking*. Just saying that it seems to be an option when you're dealing with this kind of dog and there's no other way.
*Insert Man of Steel quip here!