A nice example from classical mechanics is this: there is a hidden $SO(4)$ symmetry in the elliptical orbits of a particle in an inverse square potential, ie. the Kepler problem.
The system has an obvious $SO(3)$ symmetry because the inverse square law is invariant under rotations. But there's no a priori clue that an $SO(4)$ symmetry exists in this system.
You can read about it here: http://math.ucr.edu/home/baez/classical/runge_pro.pdf
This carries over to the quantum mechanical case when you solve the Schrödinger equation for an inverse square potential.
You can read about that here: http://hep.uchicago.edu/~rosner/p342/projs/weinberg.pdf
The result is that the hidden SO(4) symmetry explains the "coincidence" that many hydrogen atom states have the same energy.