It is often thought that possible human languages could be context-sensitive, referring to the Chomsky hierarchy. Many frameworks of formal grammar have converged on the restricted subclass of the "mildly" context-sensitive languages as a useful compromise, because the context-sensitive class itself presents problems for parsing complexity. Mildly context-sensitive languages are, by definition, parsable in polynomial time. A referee just pointed out to me something I should have remembered, which is that the universal recognition problem for the full context-sensitive class is PSPACE-complete (hugely intractable, in other words), and that Garey and Johnson's book (1979) presented an example of a single context-sensitive language that was PSPACE-complete. This means that it is less likely that a human language could be drawn from the class of context-sensitive languages beyond the "mild" ones, from a cognitive perspective.
On the other hand, I believe that the brain has quick solutions to intractable problems. More on that in a later post.