A common goal of epidemiologic research is to study how two exposures interact in causing a binary outcome. Causal interaction is defined as the presence of subjects for which the causal effect of one exposure depends on the level of the other exposure. For binary exposures, it has previously been shown that the presence of causal interaction is testable through additive statistical interaction. However, it has also been shown that the magnitude of causal interaction, defined as the proportion of subjects for which there is causal interaction, is generally not identifiable. In this article, we derive bounds on causal interactions, which are applicable to binary outcomes and categorical exposures with arbitrarily many levels. These bounds can be used to assess the magnitude of causal interaction, and serve as an important complement to the statistical test that is frequently employed. The bounds are derived both without and with an assumption about monotone exposure effects. We present an application of the bounds to a study of gene–gene interaction in rheumatoid arthritis.