  We consider ontological models of a quantum system, assuming that not all
probability distributions over the space $\Lambda$ of ontic states are
preparable, only those belonging to a certain set C. We assume further that
every POVM with a finite value space can be measured and that for every density
matrix there exists a distribution in C whose outcome statistics is given by
the density matrix. We show that this mapping from C to the set of density
matrices must be many-to-one, that is, that there must be empirically
indistinguishable distributions in C. This shows that there must be limitations
to knowledge in the sense of facts in nature that cannot be discovered
empirically.
