  In contrast with classical physics, in quantum physics some sets of
measurements are incompatible in the sense that they can not be performed
simultaneously. Among other applications, incompatibility allows for
contextuality and Bell nonlocality. This makes of crucial importance developing
tools for certifying whether a set of measurements posses a certain structure
of incompatibility. Here we show that, for quantum or nonsignaling models, if
the measurements employed in a Bell test satisfy a given type of compatibility,
then the amount of violation of some specific Bell inequalities become limited.
Then, we show that correlations arising from local measurements on two-qubit
states violate these limits, which rules out in a device-independent way such
structures of incompatibility. In particular, we prove that quantum
correlations allow for a device-independent demonstration of genuine triplewise
incompatibility. Finally, we translate these results into a
semi-device-independent Einstein-Podolsky-Rosen-steering scenario.
