Differential privacy is known to protect against threats to validity incurred due to adaptive, or exploratory, data analysis - even when the analyst adversarially searches for a statistical estimate that diverges from the true value of the quantity of interest on the underlying population. The cost of this protection is the accuracy loss incurred by differential privacy. In this work, inspired by standard models in the genomics literature, we consider data models in which individuals are represented by a sequence of attributes with the property that where distant attributes are only weakly correlated. We show that, under this assumption, it is possible to "re-use" privacy budget on different portions of the data, significantly improving accuracy without increasing the risk of overfitting.