The certification of Bell nonlocality or quantum steering implies the use of incompatible measurements. Here we make these connection quantitative. We show how to strengthen robustness-based steering and nonlocality quantifiers in order that they give strong lower bounds to previously proposed incompatibility quantifiers. Our results can be seen from two perspectives. On the one hand, it can be used to estimate how much steering or nonlocality can be demonstrated with a given set of measurements. On the other hand it gives one-sided device-independent and device independent ways of estimating measurement incompatibility.