Within the context of the prediction, detection and attribution of climate change, a number of studies have explicitly or implicitly assumed that individual climate responses to individual forcing agents can be linearly added to obtain the total climate response to the sum of the forcing agents. This assumption of the ‘linear additivity of forcing-response relationships’ has been tested by previous studies, but it remains controversial whether linear additivity holds with all combinations of forcing agents, such as ‘greenhouse gases plus indirect effects of anthropogenic aerosols’ or ‘greenhouse gases plus solar irradiance’. This study explored whether linear additivity holds in 5-year mean temperature/precipitation responses to various combinations of forcing agents in the 20th century and in a future-emissions scenario at global and continental scales. We used Model for Interdisciplinary Research on Climate version 3, which includes the first and second indirect effects of aerosols. The forcing factors considered were well-mixed greenhouse gases, the direct and indirect effects of sulphate and carbon aerosols, ozone, land-use changes, solar irradiance and volcanic aerosols (the latter three factors were specified only in the 20th-century runs). By performing and analysing an enormous matrix of forcing runs, we concluded that linear additivity holds in temperature responses for all of the combinations of forcing agents at the global and continental scales, but it breaks down for precipitation responses in certain cases of future projections.