Flux-averaged cross sections for cosmogenic-neutron activation of natural tellurium were measured using a neutron beam containing neutrons of kinetic energies up to ∼800 MeV and having an energy spectrum similar to that of cosmic-ray neutrons at sea level. Analysis of the radioisotopes produced reveals that Ag110m will be a dominant contributor to the cosmogenic-activation background in experiments searching for neutrinoless double-β decay of Te130, such as the Cryogenic Underground Observatory for Rare Events (CUORE) and the Sudbury Neutrino Observatory Plus (SNO+). An estimate of the cosmogenic-activation background in the CUORE experiment has been obtained using the results of this measurement and cross-section measurements of proton activation of tellurium. Additionally, the measured cross sections in this work are also compared with results from semiempirical cross-section calculations.