In this paper we revise a popular alternative for estimating Poisson regression models in a Bayesian framework and discuss possible pitfalls tied to data features. The MCMC algorithms are based on augmenting the model via the introduction of auxiliary variables. This leads to a model linear in the regression parameters with errors following a Gumbel or a log-Gamma distribution depending on the augmentation strategy. Such distributions are approximated by a Gaussian Mixture in order to favor standard MCMC sampling with Gibbs steps after augmentation. We show situations when such an approximation deteriorates and causes non-convergence of the algorithm, discussing how this can be detected while the algorithm is running.