This paper investigates robust fault diagnosis of multiple air data sensor faults in the presence of winds. The trade-off between robustness to winds and sensitivity to faults is challenging due to simultaneous influence of winds and latent faults on monitored sensors. Different from conventional residual generators that do not consider any constraints, we propose a constrained residual generator using moving horizon estimation. The main contribution is improved fault sensitivity by exploiting known bounds on winds in residual generation. By analyzing the Karush-Kuhn-Tucker conditions of the formulated moving horizon estimation problem, it is shown that this improvement is attributed to active inequality constraints caused by faults. When the weighting matrices in the moving horizon estimation problem are tuned to increase robustness to winds, its fault sensitivity does not simply decrease as one would expect in conventional unconstrained residual generators. Instead, its fault sensitivity increases when the fault is large enough to activate some inequality constraints. This fault sensitivity improvement is not restricted to this particular application, but can be achieved by any general moving horizon estimation based residual generator. A high-fidelity Airbus simulator is used to illustrate the advantage of our proposed approach in terms of fault sensitivity.