Polymer brushes, coatings of polymers covalently end-grafted to a surface, have been proposed as a more stable alternative to traditional physisorbed coatings. However, when such coatings are applied in settings such as vapor sensing and gas separation technologies, their responsiveness to solvent vapors becomes an important consideration. It can be anticipated that the end-anchoring in polymer brushes reduces the translational entropy of the polymers and instead introduces an entropic penalty against stretching when vapor is absorbed. Therefore, swelling can be expected to be diminished in brushes compared to nongrafted films. Here, we study the effect of the anchoring-constraint on vapor sorption in polymer coatings using coarse-grained molecular dynamics simulations as well as humidity-controlled ellipsometry on chemically identical polymer brushes and nongrafted films. We find a qualitative agreement between simulations and experiments, with both indicating that brushes certainly swell less than physisorbed films, although this effect is minor for common grafting densities. Our results imply that polymer brushes indeed hold great potential for the intended applications.