  In the case of a pair of two-outcome measurements incompatibility is
equivalent to Bell nonlocality. Indeed, any pair of incompatible two-outcome
measurements can violate the Clauser-Horne-Shimony-Holt Bell inequality, which
has been proven by Wolf et al. [Phys. Rev. Lett. 103, 230402 (2009)]. In the
case of more than two measurements the equivalence between incompatibility and
Bell nonlocality is still an open problem, though partial results have recently
been obtained. Here we show that the equivalence breaks for a special choice of
three measurements. In particular, we present a set of three incompatible
two-outcome measurements, such that if Alice measures this set, independent of
the set of measurements chosen by Bob and the state shared by them, the
resulting statistics cannot violate any Bell inequality. On the other hand,
complementing the above result, we exhibit a set of $N$ measurements for any
$N>2$ that is $(N-1)$-wise compatible, nevertheless it gives rise to Bell
violation.
