This paper introduces Bootstrapped Chains for Learning Using Privileged Information (BC-LUPI), a model-independent method designed to reconstruct privileged variables at test time. BC-LUPI builds an ensemble of predictive chains, where each chain sequentially predicts privileged variables from standard inputs and previously predicted privileged variables. The chains differ in training data via bootstrapping, and in prediction order via random permutations to provide diversity and reduce correlation between individual errors. The final output is obtained by aggregating predictions across chains which allows modeling privileged-variable dependencies and lowering variance, and, thus, improving predictive performance. Experiments on UCI datasets show that BC-LUPI is capable of outperforming baselines, particularly when the privileged information is informative and partially predictable.