This idea implies that a particle occupies every possible wavefunction it can.

This implies that a particle occupies more than one position at the same time.

I understand that no two identical particles can occupy the same energy state with the same quantum numbers.

Rather, how many particles would occupy states with energy follows as an easy consequence.

A particle occupies one point of space at each instant of time.

This shows that in Fermi-Dirac statistics, more than one particle cannot occupy a single state accessible to the system.

He also had difficulty with the assertion that a single subatomic particle can occupy numerous areas of space at one time.

The particle may only occupy certain positive energy levels.

These particles occupy "symmetric states", and can therefore share quantum states.

As a result, the charged particles can only occupy orbits with discrete energy values, called Landau levels.