Multi-center clinical trials are threatened by systematic location shifts in biomarkers measured across devices, sites, or reagent batches. Hierarchical Bayesian models are the standard remedy, borrowing strength across centers by pooling center-specific estimates. However, we prove that the standard Gaussian hierarchical posterior has a breakdown point of zero \emph{at the level of the posterior distribution itself}: a single contaminated center drives the posterior mean of every clean center to infinity and collapses the coverage of every clean center's credible interval to zero. We introduce the Posterior-Median Hierarchical Calibration (PMHC) framework, a fully Bayesian hierarchical model that replaces the Gaussian prior on center-specific shifts with a Laplace prior. Under absolute loss, the posterior median is the natural Bayes estimator, and we prove that its breakdown point is $\lfloor (K+1)/2 \rfloor$ contaminated centers---whereas the Gaussian posterior mean has breakdown point zero for every $K$. PMHC retains a full posterior distribution, enabling meaningful credible intervals and posterior probabilities. Across $K \in \{5, 10, 20\}$ centers and $n \in \{50, 200, 1000\}$ observations per center, PMHC maintains $94$--$98\%$ coverage under contamination fraction $ε= 0.60$, while Gaussian pooling collapses to $0.0\%$ coverage and its point estimate diverges by a factor of $54$. On real NHANES data ($n = 60{,}651$ across nine cycles), contamination in three cycles collapsed the Gaussian coverage to \(0.0\%\) and shifted clean cycles by \(0.41\) mg/dL, whereas PMHC maintained \(94.0\%\) coverage with a shift of \(0.008\) mg/dL. To our knowledge, this is the first explicit breakdown analysis of the \emph{posterior distribution} in hierarchical Bayesian pooling, and the first Bayesian model that is simultaneously contamination-robust and efficient in the clean-data regime.

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