edit: ^^^I was referring to a previous poster who implied that they were breeding pit bulls with border collies and just claiming the dog was a Staffy, which is a common thing to do in my area (calling pits Staffies, not the breeding). I'm not a bully breed person either but I have several friends who are, so I do know the difference and wouldn't want to offend anyone.
I have not personally seen a "Border Staffy," but I have seen a couple of accidental and one intentional mating between bully breeds and BCs. The intentional one was between an APBT and a BC, the others were between more generic pits and registered BCs. I didn't know that Border Staffies were a thing, though, I must admit, though according to Google they are! You learn something new every day.
To speak to crossbreeding, I think it's people's notions that...Dog breed A is good at this and so is Dog Breed B, so let's mix them and get...SUPER DOG!
Yeah, exactly. The instances I'm thinking about that I don't have a problem with are usually carefully considered and the dogs are somewhat similar breeds (the big one in my area is border collie/ACD crosses, and border collies were a major breed used in developing the ACD), and it's more about the individual dogs than crossing the breeds, if that makes sense. It's still a gamble in a genetic sense, but if the person knows what they're doing, it can create some cool dogs.
Usually, though, it's exactly what you describe--a kooky person who doesn't have the slightest clue about genetics trying to create a super dog, or some yahoo who decides to create a new breed for money or fame or some less obvious reason. People get some weird ideas about how genetics work.