The multivariate-t nonlinear mixed-effects model (MtNLMM) has been shown to be a promising robust tool for analyzing multiple longitudinal trajectories following arbitrary growth patterns in the presence of outliers and possible missing responses. Owing to intractable likelihood function of the model, we devise a fully Bayesian estimating procedure to account for the uncertainties of model parameters, random effects, and missing responses via the Markov chain Monte Carlo method. Posterior predictive inferences for the future values and missing responses are also investigated. We conduct a simulation study to demonstrate the feasibility of our Bayesian sampling schemes. The proposed techniques are illustrated through applications to two case studies.