In recent years, energy correlators have emerged as a powerful tool to explore the field theoretic structure of strong interactions at particle colliders. In this Letter we initiate a novel study of the nonperturbative power corrections to the projected N-point energy correlators in the limit where the angle between the detectors is small. Using the light-ray operator product expansion as a guiding principle, we derive the power corrections in terms of two nonperturbative quantities describing the fragmentation of quarks and gluons. In analogy with their perturbative leading-power counterpart, we show that power corrections obey a classical scaling behavior that is violated at the quantum level. This crucially results in a dependence on the hard scale Q of the problem that is calculable in perturbation theory. Our analytic predictions are successfully tested against Monte Carlo simulations for both lepton and hadron colliders, marking a significant step forward in the understanding of these observables.