  One of the primary goals of neutrino telescopes, such as IceCube, is the
discovery of neutrinos emitted by gamma-ray bursts (GRBs). Another source of
interest in the results obtained by these telescopes is their possible use for
tests of the applicability of Einstein's Special Relativity to neutrinos,
particularly with respect to modifications that lead to Lorentz invariance
violation that have been conjectured by some models of quantum space-time. We
examine here the fascinating scenario in which these two aspects of
neutrino-telescope physics require a combined analysis. We discuss how
neutrinos that one would not associate to a GRB, when assuming a classical
spacetime picture, may well be GRB neutrinos if the possibility that Lorentz
invariance is broken at very high energies is taken into account. As an
illustrative example we examine three IceCube high energy neutrinos that
arrived hours before GRBs (but from the same direction) and we find that the
available, IceCube data, while inconclusive, is compatible with a scenario in
which one or two of these neutrinos were GRB neutrinos and their earlier
arrival reflects Lorentz invariance violation.
  We outline how future analyses of neutrino data should be done in order to
systematically test this possibility.
