Background: Experiments designed to study rare processes, such as neutrinoless double-β decay (0νββ), are crucial tests for physics beyond the standard model. These experiments rely on reducing the intrinsic radioactive background to unprecedented levels, while adequately shielding the detectors from external sources of radioactivity. Purpose: The purpose of this work is focused on understanding the background rates from neutron interactions in Cu shielding in regions around the Q values of many candidate 0νββ decay isotopes, as well as providing data for benchmarking Monte Carlo simulations of background events. Methods: Using the broad-spectrum neutron beam at Los Alamos Neutron Science Center, we have measured γ rays emitted from inelastic neutron scattering on nat Cu. Results: We extracted the level cross sections from the γ -production cross section for 46 energy levels in nat Cu. These level cross sections were compared with the available experimental data, as well as the ENDF/B-VII evaluation for discrete levels. Conclusions: For energy levels above 2 MeV we found significant discrepancies between the suggested level cross sections for both nuclei and our data. We found reasonable agreement between our measurement and the ENDF/B-VII evaluation for the total neutron inelastic cross section in 63 Cu. Our measurement of the total neutron inelastic scattering cross section in 65 Cu was 30% lower than the ENDF/B-VII evaluations, which we attribute to unobserved transitions in 65 Cu. Furthermore, we found that the implementation of the ENDF/B-VII evaluation in simulations did not properly model the decay properties of the nucleus to the degree necessary for estimating backgrounds in rear-event searches. Finally, we examined the potential implications of our measurements on 0νββ measurements and found that many of the commonly studied 0νββ isotopes had Q values below the cutoff for ENDF/B-VII evaluated discrete levels in either Cu nucleus.