  Measurements in quantum theory can fail to be jointly measurable. Like
entanglement, this incompatibility of measurements is necessary but not
sufficient for violating Bell inequalities. The (in)compatibility relations
among a set of measurements can be represented by a joint measurability
structure, i.e., a hypergraph whose vertices denote measurements and hyperedges
denote all and only compatible sets of measurements. Since incompatibility is
necessary for a Bell violation, the joint measurability structure on each wing
of a Bell experiment must necessarily be non-trivial, i.e., it must admit a
subset of incompatible vertices. Here we show that for any non-trivial joint
measurability structure with a finite set of vertices, there exists a quantum
realization with a set of measurements that enables a Bell violation, i.e.,
given that Alice has access to this incompatible set of measurements, there
exists a set of measurements for Bob and an entangled state shared between them
such that they can jointly violate a Bell inequality. Hence, a non-trivial
joint measurability structure is not only necessary for a Bell violation, but
also sufficient. We also provide a partial characterization of qubit
measurements that are useful for Bell inequality violations in the simplest
joint measurability structure of interest, i.e., Specker's scenario, which
consists of three pairwise compatible but triplewise incompatible measurements.
