Summary Consider the linear model where one is interested in learning about β given data on y and x and when y is interval measured; that is, we observe such that . Moment inequality procedures use the implication . As compared to least squares in the classical regression model, estimates obtained using an objective function based on these moment inequalities do not provide a clear approximation to the underlying unobserved conditional mean function. Most importantly, under misspecification, it is not unusual that no parameter β satisfies the previous inequalities for all values of x, and hence minima of an objective function based on these moment inequalities are typically tight. We construct set estimates for β in the linear model that have a clear interpretation when the model is misspecified. These sets are based on moment equality models. We illustrate these sets and compare them to estimates obtained using moment inequality-based methods. In addition to the linear model with interval outcomes we also analyse the binary missing data model with a monotone instrument assumption (MIV), we find there that when this assumption is misspecified, bounds can still be non-empty, and can differ from parameters obtained via maximum likelihood. We also examine a bivariate discrete game with multiple equilibria. In sum, misspecification in moment inequality models is of a different flavour than in moment equality models, and so care should be taken with (1) the_interpretation of the estimates and (2) the size of the ‘identified set’.