We investigate several Pb$(n,{n}^{'}\ensuremath{\gamma}$) and Ge$(n,{n}^{'}\ensuremath{\gamma}$) reactions. We measure \ensuremath{\gamma}-ray production from Pb$(n,{n}^{'}\ensuremath{\gamma}$) reactions that can be a significant background for double-\ensuremath{\beta} decay experiments which use lead as a massive inner shield. Particularly worrisome for Ge-based double-\ensuremath{\beta} decay experiments are the 2041-keV and 3062-keV \ensuremath{\gamma} rays produced via Pb$(n,{n}^{'}\ensuremath{\gamma}$). The former is very close to the $^{76}\mathrm{Ge}$ double-\ensuremath{\beta} decay endpoint energy and the latter has a double escape peak energy near the endpoint. We discuss the implications of these \ensuremath{\gamma} rays on past and future double-\ensuremath{\beta} decay experiments and estimate the cross section to excite the level that produces the 3062-keV \ensuremath{\gamma} ray. Excitation \ensuremath{\gamma}-ray lines from Ge$(n,{n}^{'}\ensuremath{\gamma}$) reactions are also observed. We consider the contribution of such backgrounds and their impact on the sensitivity of next-generation searches for neutrinoless double-\ensuremath{\beta} decay using enriched germanium detectors.