Polymer-nanoparticle (NP) interfacial interactions are expected to strongly influence the properties of nanocomposites, but surprisingly, experiments often report small or no changes in the glass transition temperature, Tg. To understand this paradoxical situation, we simulate nanocomposites over a broad range of polymer-NP interaction strengths, ε. When ε is stronger than the polymer-polymer interaction, a distinct relaxation that is slower than the main α-relaxation emerges, arising from an adsorbed "bound" polymer layer near the NP surface. This bound layer "cloaks" the NPs, so that the dynamics of the matrix polymer are largely unaffected. Consequently, Tg defined from the temperature dependence of the routinely measured thermodynamics or the polymer matrix relaxation is nearly independent of ε, in accord with many experiments. Apparently, quasi-thermodynamic measurements do not reliably reflect dynamical changes in the bound layer, which alter the overall composite dynamics. These findings clarify the relation between quasi-thermodynamic Tg measurements and nanocomposite dynamics, and should also apply to thin polymer films.