No, there are limitations to the S-vN theorem. More precisely, we are led to believe that, just because it works with Q and P as a system of imprimitivities on the whole real line, then any Q and P under any assumptions will do the same. But the power of this theorem relies in the proper construction of the hypotheses which are almost never spelled out completely.
More precisely, if we consider a particle in a 1D box (the famous non-trivial problem with the rôle of the boundary conditions in determining whether the Hamiltonian and momentum operators are merely symmetric or may have self-adjoint extensions), then one can build a linear, self-adjoint operator (the time of arrival), canonically conjugate to the self-adjoint Hamiltonian operator with bounded and discrete spectrum. This operator is built in post# 8. The commutation relation, however, only holds in a domain not invariant under the action of the operators.
Here is where the mathematical finesse occurs: the S-vN theorem naturally leads to the Schwartz test function space in L^2 as a commutator domain. This has the nice property of being a common dense everywhere domain for the Q, P, Q^n and P^n. But, to be fair, the formulation of the S-vN theorem should be more „loose”, i.e. invariance of the commutator's domain be dropped, hence the construction of that time operator on a common not dense-everywhere domain becomes possible.
P.S. Why are we typically led to the deluding Schwartz space? Because of the heuristic approach to the CCR by Hermann Weyl which inspired JvN. The existence (in a strong topology) of infinite powers of Q and P by vN's work requires the Schwartz space. Therefore, von Neumann proved a stronger result, that in terms of operator series. But at the price of requiring a certain commutator domain. One of the assumptions of the famous Dixmier theorem is also the "stability of the commutator's domain". See below.
