The possibility of experimentally testing the Bell-Kochen-Specker theorem is investigated critically, following the demonstrations by Meyer, Kent, and Clifton-Kent that the predictions of quantum mechanics are indistinguishable (up to arbitrary precision) from those of a non-contextual model, and the subsequent debate about the extent to which these models are actually classical or non-contextual. The present analysis starts from a careful consideration of these ‘finite-precision’ approximations. A stronger condition for non-contextual models, dubbed ontological faithfulness, is exhibited. It is shown that this allows us to approximately formulate the constraints in Bell-Kochen-Specker theorems, such as to render the usual proofs robust. Consequently, one can experimentally test to finite precision ontologically faithful non-contextuality, and thus experimentally refute explanations from this smaller class. We include a discussion of the relation of ontological faithfulness to other proposals to overcome the finite precision objection.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘50 years of Bell’s theorem’.