When you solve Schrödinger equation for a free particle with no boundary conditions your eigen states are indexed by quantum number $k \in \mathbb R $ and $\mathbb R$ isn't countable but if you add a harmonic oscillator potential, your eigen states are indexed by harmonic number $n \in \mathbb N$ which is countable.
More interestingly for the hydrogen atom, your potential is finite as it goes to $\infty$. So for $E>0$ you have an uncountable number of states but for $E<0$ there are only a countable number of eigenstates/values.
I can't think of any counter example but I have no idea how to go about proving the number of eigenstates for a general bound system is countable.
I find this interesting because if you ignore your ability to measure position and momentum it would appear that you can change the dimensionality of your system by changing your potential.