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 inequality(1). 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 schemes(3-5), strengthen Bell's wish and make it even more timely. In 2003, Leggett proposed an alternative model for non-local correlations(6) 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 inequalities(6-9). 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.