In this paper, we study set-identified linear models. We first exhibit examples of set identification where the usual moment conditions are incomplete because they depend on a unknown bounded scalar function. We then show that incomplete linear moment conditions generate set identification where the identified set is bounded and convex. We characterize the identified set in both cases where the number of moment conditions is equal to, or greater than the number of parameters of interest. We derive consistent and asymptotically normal estimators of the support functions of these bounded and convex sets. We also construct procedures testing the validity of overidentifying restrictions which generalize the Sargan test. Some empirical illustrations on income data and on artificial data are provided.