  The observation of quantum nonlocality, i.e. quantum correlations violating a
Bell inequality, implies the use of incompatible local quantum measurements.
Here we consider the converse question. That is, can any set of incompatible
measurements be used in order to demonstrate Bell inequality violation? Our
main result is to construct a local hidden variable model for an incompatible
set of qubit measurements. Specifically, we show that if Alice uses this set of
measurements, then for any possible shared entangled state, and any possible
dichotomic measurements performed by Bob, the resulting statistics are local.
This represents significant progress towards proving that measurement
incompatibility does not imply Bell nonlocality in general.
