Bivariate clustered (correlated) data often encountered in epidemiological and clinical research are routinely analyzed under a linear mixed model (LMM) framework with underlying normality assumptions of the random effects and within-subject errors. However, such normality assumptions might be questionable if the data set particularly exhibits skewness and heavy tails. Using a Bayesian paradigm, we use the skew-normal/independent (SNI) distribution as a tool for modeling clustered data with bivariate non-normal responses in an LMM framework. The SNI distribution is an attractive class of asymmetric thick-tailed parametric structure which includes the skew-normal distribution as a special case. We assume that the random effects follow multivariate SNI distributions and the random errors follow SNI distributions which provides substantial robustness over the symmetric normal process in an LMM framework. Specific distributions obtained as special cases, viz. the skew-t, the skew-slash and the skew-contaminated normal distributions are compared, along with the default skew-normal density. The methodology is illustrated through an application to a real data which records the periodontal health status of an interesting population using periodontal pocket depth (PPD) and clinical attachment level (CAL). Copyright © 2010 John Wiley & Sons, Ltd.