  Quantum theory predicts and experiments confirm that nature can produce
correlations between distant events that are nonlocal in the sense of violating
a Bell inequality. Nevertheless, Bell's strong sentence {\it Correlations cry
out for explanations} remains relevant. The maturing of quantum information
science and the discovery of the power of nonlocal correlations, e.g. for
cryptographic key distribution beyond the standard Quantum Key Distribution
schemes, strengthen Bell's wish and make it even more timely.
  In 2003, Leggett proposed an alternative model for nonlocal correlations
[Found. Phys. 33, 1469 (2003)], 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. Remarkably 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 Leggett-type inequalities. The
simplest of these inequalities is experimentally violated. Then we go beyond
Leggett's model, and show that one cannot ascribe even partially defined
individual properties to the components of a maximally entangled pair.
