  Weak value measurements have recently given rise to a large interest for both
the possibility of measurement amplification and the chance of further quantum
mechanics foundations investigation. In particular, a question emerged about
weak values being proof of the incompatibility between Quantum Mechanics and
Non-Contextual Hidden Variables Theories (NCHVT). A test to provide a
conclusive answer to this question was given in [M. Pusey, Phys. Rev. Lett.
113, 200401 (2014)], where a theorem was derived showing the NCHVT
incompatibility with the observation of anomalous weak values under specific
conditions. In this paper we realize this proposal, clearly pointing out the
strict connection between weak values and the contextual nature of Quantum
Mechanics.
