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We study the sensitivity and practicality of Hendersonâ€™s theorem in classical statistical mechanics, which states that the pair potential \(v(r)\) that gives rise to a given pair correlation function \(g_2(r)\) [or equivalently, the structure factor \(S(k)\)] in a classical many-body system at number density \(\rho\) and temperature \(T\) is unique up to an additive constant. While widely invoked in inverse-problem studies, the utility of the theorem has not been quantitatively scrutinized to any large degree. We show that Hendersonâ€™s theorem has practical shortcomings for disordered and ordered phases for certain densities and temperatures. Using proposed sensitivity metrics, we identify illustrative cases in which distinctly different potential functions give very similar pair correlation functions and/or structure factors up to their corresponding correlation lengths. Our results reveal that due to a limited range and precision of pair information in either direct or reciprocal space, there is effective ambiguity of solutions to inverse problems that utilize pair information only, and more caution must be exercised when one claims the uniqueness of any resulting effective pair potential found in practice. We have also identified systems that possess virtually identical pair statistics but have distinctly different higher-order correlations. Such differences should be reflected in their individually distinct dynamics (e.g., glassy behaviors). Finally, we prove a more general version of Hendersonâ€™s theorem that extends the uniqueness statement to include potentials that involve two- and higher-body interactions.