  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.
