Recently, we introduced an approach for more easily interpreting searches for resonances at the LHC - and to aid in distinguishing between realistic and unrealistic alternatives for potential signals. This `simplfied limits' approach was derived using the narrow width approximation (NWA) - and therefore was not obviously relevant in the case of wider resonances. Here, we broaden the scope of the analysis. First, we explicitly generalize the formalism to encompass resonances of finite width. We then examine how the width of the resonance modifies bounds on new resonances that are extracted from LHC searches. Second, we demonstrate, using a wide variety of cases, with different incoming partons, resonance properties, and decay signatures, that the limits derived in the NWA yield pertinant, and somewhat conservative (less stringent) bounds on the model parameters. We conclude that the original simplified limits approach is useful in the early stages of evaluating and interpreting new collider data and that the generalized approach is a valuable further aid when evidence points toward a broader resonance.

26 pages, 6 figures (figures improved, minor additions to text for publication)