Abstract : For diseases with more than one risk factor, the sum of probabilistic estimates of the number of cases attributable to each individual factor may exceed the total number of cases observed, especially when uncertainties about exposure and dose-response for some risk factors is high. In this study we outline a method to bound the fraction of lung cancer fatalities not attributed to specific well-studied causes. Such information serves as a "reality check" for attributional studies of the minor risk factors, and, as such, complements the traditional risk analysis. With lung cancer as our example, we attribute portions of the observed lung cancer mortality to known causes (such as smoking, residential radon, and asbestos fibers) and describe the uncertainty surrounding those estimates. The interactions among the risk factors are also quantified, to the extent possible. We then infer an upper bound on the residual risk due to "other" causes, using a coherence constraint on the total number of deaths, the maximum uncertainty principle, and the mathematics of imprecise probabilities.