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→∞) for all compressions Δϕ studied, and at jamming onset Δϕ→0 for all N. We perform fixed energy simulations following perturbations with amplitude δ along eigendirections of the dynamical matrix. The fluctuations abruptly spread to all modes for δ≈δ(c) (where a single contact breaks) in contrast to linear and weakly nonlinear behavior. For δ > δ(c), all discrete modes disappear into a continuous frequency band. <δ(c)> scales with 1/N and Δϕ, 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.