Is it known that there are provable theorems out there that are hopelessly beyond the human intellect to prove? I've never heard that claim made before in quite the way you just made it. Of course, there are a whole host of things that are undecidable in Godel's sense of it, but that's a fundamental constraint having nothing to do with our intelligence or lack of it. I've often thought that perhaps there is some "critical mass" ito intelligence, and that once a species exceeds that critical mass, it can in principle prove anything that can be proven or, alternatively, understand any valid proof. As Albert Einstein was supposed to have said, "The most incomprehensible thing about the universe is that it is comprehensible." Hope that's true. It's gonna be depressing if we're just another species of dumb apes.
Have you looked at some of Chaitin's work? The digits of his Omega number are an interesting case. His take on it is interesting: anything sufficiently complex is essentially random.
But on a more mundane level, any finite theorem proving machine (and I'm assuming people are that) will be helpless in the face of a theorem whose smallest proof exceeds the capacity of the machine.