Social scientists are often interested in testing multiple causal mechanisms through which a treatment affects outcomes. A predominant approach has been to use linear structural equation models and examine the statistical significance of the corresponding path coefficients. However, this approach implicitly assumes that the multiple mechanisms are causally independent of one another. In this article, we consider a set of alternative assumptions that are sufficient to identify the average causal mediation effects when multiple, causally related mediators exist. We develop a new sensitivity analysis for examining the robustness of empirical findings to the potential violation of a key identification assumption. We apply the proposed methods to three political psychology experiments, which examine alternative causal pathways between media framing and public opinion. Our analysis reveals that the validity of original conclusions is highly reliant on the assumed independence of alternative causal mechanisms, highlighting the importance of proposed sensitivity analysis. All of the proposed methods can be implemented via an open source R package, mediation.