I was browsing MO and I have stumbled upon this answer which discusses why we should expect homotopy groups of spheres to be complicated. One heuristic argument given is that "the theory needs to have some complexity somewhere", which means that since spheres have simple cohomology, they have to have complicated homotopy, and dually for EM spaces, since they have simple homotopy, they have complicated cohomology.
Now, I understand that this might have been nothing more than a heuristic given to prove a point -- after all, they are spaces with both homotopy and cohomology simple, like the contractible ones. However, something tells me that there is something actually going on there. Hence the question:
Is there some sense in which a (sufficiently nontrivial) space cannot have both homotopy and cohomology simple?
If this is not a general phenomenon, what are some good (families of) examples for which both homotopy and cohomology are simple?