Cosmic rays are the highest energy particles available for our study and as such serve as excellent probes of the effects of Lorentz Invariance Violations, which are expected to increase with energy. This general paradigm is investigated in this paper by studying the effects of such violations within the Coleman-Glashow model in which each particle species may have its own maximum attainable velocity, even exceeding that of light \textit{in vacuo}. The particular focus here is that the muon neutrino may have the maximum speed exceeding that of light. We show that such an assumption leads to the elongation of the decay lifetime of the pion that increases with energy over and above the time dilation effects. We provide a transparent analytical derivation of the spectral intensities of muon neutrinos and muons generated in the Earth's atmosphere by cosmic rays. In this derivation we not only account for elongation of the pion lifetime, but also for the loss of energy by the neutrinos by radiation of the electron-positron pairs through the Cohen-Glashow process, during their propagation. We then compare the theoretical spectra with observations of neutrinos and muons from large instruments like IceCube and BUST to set a limit of $\sim10^{-13}$ on the fractional excess speed of neutrinos over that of light. We also show that the ratio of the spectral intensities of downward and upward moving neutrinos at various angles constitute a diagnostic exclusively for the Cohen-Glashow process, which may be searched for in the IceCube data set. We conclude the paper with several comments, including those related to improvements of these tests when definite signals of GZK neutrinos will be observed.

version 2, 28 pages, 10 figures