A good example is the problem of the orbit of an electron around the nucleus of an atom.

Everyone knew the relative sizes of the nucleus and the electron and had assumed a solar system model i.e. the electron revolved around the nucleus as the earth revolved round the sun.

The problem was - as the electron is losing energy, why did n't it spiral into the nucleus just as a satellite will spiral in to the Earth .

The best physicists in the world spent many years trying to solve this and could not.
Until a young Danish student called Niels Bohr did. Apocryphally, he is to have said

Let us abandon Classical Physics

This is a tremendous statement but once it was said, a new physics was invented, Quantum Physics and the problem is then solved in three lines.

It wasn't the problem that was difficult but we were using the wrong method to describe it.