The theory of constrained best approximation in Hilbert spaces has been systematically developed well over a decade and effective characterizations of best approximations have been known under some qualifications on the constraints. Yet, the existing theory does not explain how to characterize a best approximation in the face of data uncertainty in the constraints, despite the reality that the data of the constraints are often uncertain (that is, they are not known exactly) due to estimation errors, prediction errors or lack of information. This paper explains when the best approximation over uncertain linear constraints in a real Hilbert space is immunized against bounded data uncertainty. This study is done by characterizing the best approximation of the robust counterpart of the uncertain constrained best approximation problem where the constraints are enforced for all possible uncertainties within the prescribed uncertainty sets. We show that under a new robust strong conical hull intersection property (robust strong CHIP) the same kind of effective characterizations of constrained best approximation hold for the robust best approximation that is immunized against bounded data uncertainty. We also establish a strong duality theorem for the robust constrained best approximation problem and its associated dual problem under the robust strong CHIP. Some examples are given to illustrate the obtained results.