Quantum theory predicts and experiments confirm that nature can produce correlations between distant events that are non-local in the sense of violating a Bell inequality1. Nevertheless, Bell's strong sentence 'Correlations cry out for explanations' (ref. 2) remains relevant. The maturing of quantum information science and the discovery of the power of non-local correlations, for example for cryptographic key distribution beyond the standard quantum key distribution schemes3, 4, 5, strengthen Bell's wish and make it even more timely. In 2003, Leggett proposed an alternative model for non-local correlations6 that he proved to be incompatible with quantum predictions. We present here a new approach to this model, along with new inequalities for testing it. These inequalities can be derived in a very simple way, assuming only the non-negativity of probability distributions; they are also stronger than previously published and experimentally tested Leggett-type inequalities6, 7, 8, 9. The simplest of the new inequalities is experimentally violated. Then we go beyond Leggett's model, and show that we cannot ascribe even partially defined individual properties to the components of a maximally entangled pair.