Incoherent neutron scattering measurements were performed on polycarbonate (PC), poly(methyl methacrylate) (PMMA), and poly(vinyl chloride) (PVC) films of variable thickness, ranging from bulklike down to 75 Angstrom, or length scales comparable to the polymer's radius of gyration. The temperature dependencies of the incoherent elastic scattering are analyzed in terms of a Debye-Waller factor to estimate the hydrogen-weighted mean-square atomic displacement .We find a general reduction of as the polymer films become increasingly thin, especially above the calorimetric glass transition temperature, T, However, below T-g this reduction depends strongly upon the relative amplitude of the displacement. Specifically, if in the bulk glass is especially large, as seen in PC, the extensive sub-T-g motions are strongly suppressed by thin film confinement. On the contrary, glassy PVC displays comparatively small-amplitude displacements in the glass and virtually no reduction of upon confinement. These results are discussed in terms of a caging of the atomic motions as the degree of thin film confinement increases.