The emergence of big data has led to so-called convergence complexity analysis, which is the study of how Markov chain Monte Carlo (MCMC) algorithms behave as the sample size, $n$, and/or the number of parameters, $p$, in the underlying data set increase. This type of analysis is often quite challenging, in part because existing results for fixed $n$ and $p$ are simply not sharp enough to yield good asymptotic results. One of the first convergence complexity results for an MCMC algorithm on a continuous state space is due to Yang and Rosenthal (2019), who established a mixing time result for a Gibbs sampler (for a simple Bayesian random effects model) that was introduced and studied by Rosenthal (1996). The asymptotic behavior of the spectral gap of this Gibbs sampler is, however, still unknown. We use a recently developed simulation technique (Qin et. al., 2019) to provide substantial numerical evidence that the gap is bounded away from 0 as $n \rightarrow \infty$. We also establish a pair of rigorous convergence complexity results for two different Gibbs samplers associated with a generalization of the random effects model considered by Rosenthal (1996). Our results show that, under strong regularity conditions, the spectral gaps of these Gibbs samplers converge to 1 as the sample size increases.