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. A precise quantification characterizing the role of the dimension as well as the dependence is provided. This surprising behavior has serious implications towards 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 at parts of the true function of interest. Without further modification, we show that truncated normal priors are not suitable for modeling at regions and propose a novel alternative strategy based on shrinking the coordinates using a multiplicative scale parameter. The proposed shrinkage prior is empirically shown to guard against the mass shifting phenomenon while retaining computational efficiency.