We show that lower-dimensional marginal densities of dependent zero-mean normal distributions truncated to the positive orthant exhibit a mass-shifting phenomenon. Despite the truncated multivariate normal density having a mode at the origin, the marginal density assigns increasingly small mass near the origin as the dimension increases. The phenomenon accentuates with stronger correlation between the random variables. This surprising behavior has serious implications toward Bayesian constrained estimation and inference, where the prior, in addition to having a full support, is required to assign a substantial probability near the origin to capture flat parts of the true function of interest. A precise quantification of the mass-shifting phenomenon for both the prior and the posterior, characterizing the role of the dimension as well as the dependence, is provided under a variety of correlation structures. Without further modification, we show that truncated normal priors are not suitable for modeling flat regions and propose a novel alternative strategy based on shrinking the coordinates using a multiplicative scale parameter. The proposed shrinkage prior is shown to achieve optimal posterior contraction around true functions with potentially flat regions. Synthetic and real data studies demonstrate how the modification guards against the mass shifting phenomenon while retaining computational efficiency. Supplementary materials for this article are available online.