Realistic models of particle physics include many scalar fields. These fields generically have nonminimal couplings to the Ricci curvature scalar, either as part of a generalized Einstein theory or as necessary counterterms for renormalization in curved background spacetimes. We develop a gauge-invariant formalism for calculating primordial perturbations in models with multiple nonminimally coupled fields. We work in the Jordan frame (in which the nonminimal couplings remain explicit) and identify two distinct sources of entropy perturbations for such models. One set of entropy perturbations arises from interactions among the multiple fields. The second set arises from the presence of nonminimal couplings. Neither of these varieties of entropy perturbations will necessarily be suppressed in the long-wavelength limit, and hence they can amplify the curvature perturbation zeta, even for modes that have crossed outside the Hubble radius. Models that overproduce long-wavelength entropy perturbations endanger the close fit between predicted inflationary spectra and empirical observations.