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.
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