While for finite potentials in one dimension there is only one state
for each allowed energy, it is quite common to find multiple states
with the same energy for potentials in higher dimensions which possess
a high degree of symmetry. The existence of two independent states
with the same energy is referred to as a degeneracy. We say that the
three dimensional SHO potential function 
is spherically symmetric
because the value of the potential depends only on distance
from the origin. Because of this high
degree of symmetry, the states of the three dimension SHO are are
highly degenerate, many states share the same energy.