Many jammed particulate systems, such as granular and colloidal materials, interact via repulsive contact forces. We find that these systems possess no harmonic regime in the large system limit (N -> infinity) for all compressions Delta phi studied, and at jamming onset Delta phi -> 0 for all N. We perform fixed energy simulations following perturbations with amplitude delta along eigendirections of the dynamical matrix. The fluctuations abruptly spread to all modes for delta approximate to delta(c) (where a single contact breaks) in contrast to linear and weakly nonlinear behavior. For delta > delta(c), all discrete modes disappear into a continuous frequency band. scales with 1/N and Delta phi, which limits harmonic behavior to only overcompressed systems. The density of vibrational modes deviates strongly from that predicted from the dynamical matrix when the system enters the nonharmonic regime, which significantly affects its mechanical and transport properties.