  Recent time-of-flight measurements on muon neutrinos in the OPERA neutrino
oscillation experiment have found anomalously short times compared to the light
travel-times, corresponding to a superluminal velocity, $v-1=2.37\pm0.32\times
10^{-5}$ in units where $c=1$. We show that cosmological bounds rule out an
explanation involving a Lorentz invariant tachyonic neutrino. At the OPERA
energy scale, nucleosynthesis constraints imply $v-1<0.86\times 10^{-12}$ and
the Cosmic Microwave Background observations imply $v-1<7.1\times 10^{-23}$.
The CMB limit on the velocity of a tachyon with an energy of 10 MeV is stronger
than the SN1987A limit. Superluminal neutrinos that could be observed at
particle accelerator energy scales would have to be associated with Lorentz
symmetry violation.
