The ill-posedness of the nonparametric instrumental variable (NPIV) model leads to estimators that may suffer from poor statistical performance. In this paper, we explore the possibility of imposing shape restrictions to improve the performance of the NPIV estimators. We assume that the function to be estimated is monotone and consider a sieve estimator that enforces this monotonicity constraint. We define a constrained measure of ill-posedness that is relevant for the constrained estimator and show that, under a monotone IV assumption and certain other mild regularity conditions, this measure is bounded uniformly over the dimension of the sieve space. This finding is in stark contrast to the well known result that the unconstrained sieve measure of ill-posedness that is relevant for the unconstrained estimator grows to innity with the dimension of the sieve space. Based on this result, we derive a novel non-asymptotic error bound for the constrained estimator. The bound gives a set of data-generating processes for which the monotonicity constraint has a particularly strong regularization effect and considerably improves the performance of the estimator. The form of the bound implies that the regularization effect can be strong even in large samples and even if the function to be estimated is steep, particularly so if the NPIV model is severely ill-posed. Our simulation study conrms these findings and reveals the potential for large performance gains from imposing the monotonicity constraint.