  We study the extent to which \psi-epistemic models for quantum measurement
statistics---models where the quantum state does not have a real, ontic
status---can explain the indistinguishability of nonorthogonal quantum states.
This is done by comparing the overlap of any two quantum states with the
overlap of the corresponding classical probability distributions over ontic
states in a \psi-epistemic model. It is shown that in Hilbert spaces of
dimension $d \geq 4$, the ratio between the classical and quantum overlaps in
any \psi-epistemic model must be arbitrarily small for certain nonorthogonal
states, suggesting that such models are arbitrarily bad at explaining the
indistinguishability of quantum states. For dimensions $d$ = 3 and 4, we
construct explicit states and measurements that can be used experimentally to
put stringent bounds on the ratio of classical-to-quantum overlaps in
\psi-epistemic models, allowing one in particular to rule out maximally
\psi-epistemic models more efficiently than previously proposed.
