Are there any hard SAT instances that can be quickly solved?

Question

Are there any hard SAT instances that can be quickly solved?

Is there a special case?

I'm not talking about the whole SAT.

But are there any special cases of hard SAT formulas that can be solved in polynomial time?

We exclude class P resulting from Schaefer's dichotomy theorem for the SAT.

--- Edit:

I may have asked the wrong question:

Are there such special cases known? - "known" - it should be.