We obtain the maximum diffuse neutrino intensity predicted by hadronic photoproduction models of the type which have been applied to the jets of active galactic nuclei or gamma ray bursts. For this, we compare the proton and gamma ray fluxes associated with hadronic photoproduction in extragalactic neutrino sources with the present experimental upper limit on cosmic ray protons and the extragalactic gamma ray background, employing a transport calculation of energetic protons traversing cosmic photon backgrounds. We take into account the effects of the photon spectral shape in the sources on the photoproduction process, cosmological source evolution, the optical depth for cosmic ray ejection, and discuss the possible effects of magnetic fields in the vicinity of the sources. For photohadronic neutrino sources which are optically thin to the emission of neutrons we find that the cosmic ray flux imposes a stronger bound than the extragalactic gamma ray background in the energy range between 10^5 GeV and 10^11 GeV, as previously noted by Waxman & Bahcall (1999). We also determine the maximum contribution from the jets of active galactic nuclei, using constraints set to their neutron opacity by gamma-ray observations. This present upper limit is consistent with the jets of active galactic nuclei producing the extragalactic gamma ray background hadronically, but we point out future observations in the GeV-to-TeV regime could lower this limit. We also briefly discuss the contribution of gamma ray bursts to ultra-high energy cosmic rays as it can be inferred from possible observations or limits on their correlated neutrino fluxes.
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