Nonlinear mixed-effects (NLME) models are popular in many longitudinal studies, including those on human immunodeficiency virus (HIV) viral dynamics, pharmacokinetic analysis, and studies of growth and decay analysis. Generally, the normality of the random effects is a common assumption in NLME models but it can sometimes be unrealistic, suppressing important features of among-subjects variation. In this context, the use of normal/independent distributions arises as a tool for robust modeling of NLME models. These distributions fall in a class of symmetric heavy-tailed distributions that includes the normal distribution, the generalized Student-t, Student-t, slash and the contaminated normal distributions as special cases, providing an appealing robust alternative to the routine use of normal distributions in these types of models. The aim of this paper is the estimation of NLME models considering normal/independent distributions for the error term and random effects, under the Bayesian paradigm. A Bayesian case deletion influence diagnostic based on the q-divergence measure and model selections criteria is also developed. These analyses are computationally possible due to an important result that approximates the likelihood function of a NLME model with normal/independent distributions for a simple normal/independent distribution with specified parameters. An example of the new method is presented through simulation and application to a real dataset of AIDS/HIV infected patients that was initially analyzed using a normal NLME model.