Dimensionality of Hilbert space Simple question, but I can't seem to find the answer searching very easily.
Does the dimensionality of a Hilbert space correspond to the number of possible states a system can take on? ("The system" being the one the Hilbert space describes.)
 A: The dimension of a (Hilbert-)space is the number of basis vectors in any basis, i.e. the maximum number of linear independent states one can find. Since the eigenstates of any hermitian operator form a (orthogonal) basis of the space, the dimension of the (Hilbert-)space also corresponds to the number of possible outcomes of an observable, but only if counting $n$-fold degenerate eigenvalues $n$ times. In particular, if the energy levels of a system are non-degenerate, their number corresponds to the dimension.
This might have been what you meant when you said 

...dimensionality of a Hilbert space correspond to the number of possible states a system can take on?

However, keep in mind that any (normalized) superposition of basis states is also an allowed state the system can take on, it's in general just not an eigenstate of, say, energy. Thus, the number of possible states is infinite (except maybe for the case of a single level).
A: Here is a physical/information-theoretic answer to complement the existing mathematical ones. The Hilbert space dimension is the number of mutually distinguishable states that a system can be in. By saying that two states $|\psi\rangle$ and $|\phi\rangle$ are distinguishable I mean the following. If I am given a system that is either in the state $|\phi\rangle$ or the state $|\psi\rangle$, quantum mechanics permits in principle the existence of a single measurement that will tell me with complete certainty whether the system is in $|\psi\rangle$ or $|\phi\rangle$. This is only possible if $\langle \phi |\psi\rangle = 0$, so the number of mutually distinguishable states of a system is also the number of mutually orthogonal states, i.e. the Hilbert space dimension.
