We study the problem of reconstructing tabular data from aggregate statistics, in which the attacker aims to identify interesting claims about the sensitive data that can be verified with 100% certainty given the aggregates. Successful attempts in prior work have conducted studies in settings where the set of published statistics is rich enough that entire datasets can be reconstructed with certainty. In our work, we instead focus on the regime where many possible datasets match the published statistics, making it impossible to reconstruct the entire private dataset perfectly (i.e., when approaches in prior work fail). We propose the problem of partial data reconstruction, in which the goal of the adversary is to instead output a $\textit{subset}$ of rows and/or columns that are $\textit{guaranteed to be correct}$. We introduce a novel integer programming approach that first $\textbf{generates}$ a set of claims and then $\textbf{verifies}$ whether each claim holds for all possible datasets consistent with the published aggregates. We evaluate our approach on the housing-level microdata from the U.S. Decennial Census release, demonstrating that privacy violations can still persist even when information published about such data is relatively sparse.