Cosmogenic neutrinos originate from photo-hadronic interactions of cosmic ray protons with the cosmic microwave background (CMB). The neutrino production rate can be constrained through the accompanying electrons, positrons and gamma-rays that quickly cascade on the CMB and intergalactic magnetic fields to lower energies and generate a γ-ray background in the GeV–TeV region. Bethe–Heitler pair production by protons also contributes to the cascade and can tighten the neutrino constraints in models where extragalactic cosmic rays begin to dominate over the galactic component at a relatively low “crossover” energy. We investigate this issue in the light of the recent Fermi-LAT measurements of the diffuse extragalactic γ-ray background and illustrate by a fit to the HiRes spectrum how the prediction of the cosmogenic neutrino flux in all-proton models varies with the crossover energy. The neutrino flux is required to be smaller when the gamma-ray bound is applied, nevertheless such models are still consistent with HiRes and Fermi-LAT if one properly takes into account the energy uncertainty of cosmic ray measurements. The presently allowed flux is within reach of the IceCube neutrino telescope and other dedicated radio experiments.