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A Bayesian covariance structure model (BCSM) is proposed for interval-censored multi-way nested survival data. This flexible modeling framework generalizes mixed effects survival models by allowing positive and negative associations among clustered observations. Conjugate shifted-inverse gamma priors are proposed for the covariance parameters, implying inverse gamma priors for the eigenvalues of the covariance matrix, which ensures a positive definite covariance matrix under posterior analysis. A numerically efficient Gibbs sampling procedure is defined for balanced nested designs. This requires sampling latent variables from their marginal full conditional distributions, which are derived through a recursive formula. This makes the estimation procedure suitable for interval-censored data with large cluster sizes. For unbalanced nested designs, a novel (balancing) data augmentation procedure is introduced to improve the efficiency of the Gibbs sampler. The Gibbs sampling procedure is validated in two simulation studies. The linear transformation BCSM (LT-BCSM) was applied to two-way nested interval-censored event times to analyze differences in adverse events between three groups of patients, who were randomly allocated to treatment with different stents (BIO-RESORT). The parameters of the structured covariance matrix represented unobserved heterogeneity in treatment effects and were examined to detect differential treatment effects. A comparison was made with inference results under a random effects linear transformation model. It was concluded that the LT-BCSM led to inferences with higher posterior credibility, a more profound way of quantifying evidence for risk equivalence of the three treatments, and it was more robust to prior specifications. [ABSTRACT FROM AUTHOR]