Pulsar timing array (PTA) searches for a gravitational-wave background (GWB) typically include time-correlated “red” noise models intrinsic to each pulsar. Using a simple simulated PTA data set with an injected GWB signal we show that the details of the red noise models used, including the choice of amplitude priors and even which pulsars have red noise, have a striking impact on the GWB statistics, including both upper limits and estimates of the GWB amplitude. We find that the standard use of uniform priors on the red noise amplitude leads to 95% upper limits, as calculated from one-sided Bayesian credible intervals, that are less than the injected GWB amplitude 50% of the time. In addition, amplitude estimates of the GWB are systematically lower than the injected value by 10%-40%, depending on which models and priors are chosen for the intrinsic red noise. We tally the effects of model and prior choice and demonstrate how a “dropout” model, which allows flexible use of red noise models in a Bayesian approach, can improve GWB estimates throughout.