Hiring processes optimize for people who are good at getting hired. Working with someone (if you're willing to prune the bad coworkers) optimizes for people who are good colleagues.
In those circumstances, when yes a fundamental mistake has been made, I believe that in theory it's best to end the relationship quickly, for both parties involved. In practice it's obviously not that easy or simple.
I've heard of companies instituting mandatory work days or even work weeks for prospective hires to vet how they will fit into the workplace. That sort of thing is attempting to solve the issue but how on earth do you convince anyone, especially someone already gainfully employed, to go through with that process? I suppose I'd do it for a company I absolutely, positively, had to go work for (very few if any)... realistically it seems hard to accomplish if not rude to request.
Ultimately I think the hiring process is one in which mistakes can happen. What's the most equitable way to resolve them when they do?