What does an experimental test of quantum contextuality prove or disprove?

The possibility to test experimentally 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 to which extent these models are actually classical or non-contextual. The present analysis starts from a careful consideration these "finite-precision" approximations. A stronger condition for non-contextual models, dubbed <ontological faithfulness>, is exhibited. It is shown that it allows to formulate approximately the constraints in Bell-Kochen-Specker theorems such as to render the usual proofs robust. As a consequence, 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.