There may be a high-energy cutoff of neutrino events in IceCube data. In particular, IceCube does not observe either continuum events above 2PeV, or the Standard Model Glashow-resonance events expected at 6.3PeV. There are also no higher energy neutrino signatures in the ANITA and Auger experiments. This absence of high-energy neutrino events motivates a fundamental restriction on neutrino energies above a few PeV. We postulate a simple scenario to terminate the neutrino spectrum that is Lorentz-invariance violating, but with a limiting neutrino velocity that is always smaller than the speed of light. If the limiting velocity of the neutrino applies also to its associated charged lepton, then a significant consequence is that the two-body decay modes of the charged pion are forbidden above two times the maximum neutrino energy, while the radiative decay modes are suppressed at higher energies. Such stabilized pions may serve as cosmic ray primaries.