What is known about the existence of other pairs of spheres (such as $S^2$ and $S^3$) whose homotopy groups coincide starting from some index.

A sufficient condition for this is the existence of a fiber bundle $S^m \to S^n$ with fiber having a finite number of nonzero homotopy groups (as in the case of the Hopf fibration)

P.S. I don't know if my question has a research level. If it is not, then feel free to close it.