Many detection and attribution and pattern scaling studies assume that the global climate response to multiple forcings is additive: that the response over the historical period is statistically indistinguishable from the sum of the responses to individual forcings. Here, we use the NASA Goddard Institute for Space Studies (GISS) and National Center for Atmospheric Research Community Climate System Model (CCSM4) simulations from the CMIP5 archive to test this assumption for multi-year trends in global-average, annual-average temperature and precipitation at multiple timescales. We find that responses in models forced by pre-computed aerosol and ozone concentrations are generally additive across forcings. However, we demonstrate that there are significant nonlinearities in precipitation responses to different forcings in a configuration of the GISS model that interactively computes these concentrations from precursor emissions. We attribute these to differences in ozone forcing arising from interactions between forcing agents. Our results suggest that attribution to specific forcings may be complicated in a model with fully interactive chemistry and may provide motivation for other modeling groups to conduct further single-forcing experiments.