even hypothetically this is only partially possible. for sure what you talk about would be something that is purely an abstract / logical problem, because if something we effectively "can't think" originates somehow in the real world, we would be able to observe it and realize that we can't reason about it.

and even when we have exhausted everything there is to know in the universe and solved every type of problem, we still can't be sure that there isn't a magic gap that we can't see using our intelligence.

but this is completely a god of the gaps argument, i.e. a religious argument, and thus not worth considering until we get actual evidence of such a thing.