Abstract The diffuse flux of cosmic neutrinos has been measured by the IceCube Observatory from TeV to PeV energies. We show that an improved characterization of this flux at lower energies, TeV and sub-TeV, reveals important information on the nature of the astrophysical neutrino sources in a model-independent way. Most significantly, it could confirm the present indications that neutrinos originate in cosmic environments that are optically thick to GeV–TeV γ -rays. This conclusion will become inevitable if an uninterrupted or even steeper neutrino power law is observed in the TeV region. In such γ -ray-obscured sources, the γ -rays that inevitably accompany cosmic neutrinos will cascade down to MeV–GeV energies. The requirement that the cascaded γ -ray flux accompanying cosmic neutrinos should not exceed the observed diffuse γ -ray background puts constraints on the peak energy and density of the radiation fields in the sources. Our calculations inspired by the existing data suggest that a fraction of the observed diffuse MeV–GeV γ -ray background may be contributed by neutrino sources with intense radiation fields that obscure the high-energy γ -ray emission accompanying the neutrinos.