Copulas have enjoyed increased usage in many areas of econometrics, including applications with discrete outcomes. However, Genest and Nešlehová (2007) present evidence that copulas for discrete outcomes are not identified, particularly when those discrete outcomes follow count distributions. This paper confirms the Genest and Nešlehová result using a series of simulation exercises. The paper then proceeds to show that those identification concerns diminish if the model has a regression structure such that the exogenous variable(s) generates additional variation in the outcomes and thus more completely covers the outcome domain.