In this paper, we propose a methodology which a) evaluates the effect of covariates on doubly interval-censored paired responses, b) is based on minimal parametric assumptions concerning the distributional parts of the model and c) evaluates the association between the two responses of the pair. Our methodology tackles three research questions arising from the Signal Tandillobiel (R) project, a prospective Flemish (Belgian) longitudinal dental study. The research questions are 1) What is the effect of baseline covariates on the time-to-caries of the permanent right first molars? 2) Is the effect of the covariates the same for the upper and lower teeth? 3) What is the association between the times-to-caries on the upper and lower teeth? Time-to-caries is defined as the difference of two interval-censored observations, caries time and emergence time, and hence it is a doubly interval-censored response. We Suggest using an accelcrated failure time model with a bivariate smooth error distribution being a mixture of bivariate normal components defined on a fine fixed grid. To deal with the problem of doubly interval censoring, we use Bayesian methodology and Markov chain Monte Carlo sampling.