We present accurate mass and thermodynamic profiles for 57 galaxy clusters observed with the Chandra X-ray Observatory. We investigate the effects of local gravitational acceleration in central cluster galaxies, and explore the role of the local free-fall time (t(ff)) in thermally unstable cooling. We find that the radially averaged cooling time (t(cool)) is as effective an indicator of cold gas, traced through its nebular emission, as the ratio t(cool)/t(ff). Therefore, t(cool) primarily governs the onset of thermally unstable cooling in hot atmospheres. The location of the minimum t(cool)/t(ff), a thermodynamic parameter that many simulations suggest is key in driving thermal instability, is unresolved in most systems. Consequently, selection effects bias the value and reduce the observed range in measured t(cool)/t(ff) minima. The entropy profiles of cool-core clusters are characterized by broken power laws down to our resolution limit, with no indication of isentropic cores. We show, for the first time, that mass isothermality and the K proportional to r(2/3) entropy profile slope imply a floor in t(cool)/t(ff) profiles within central galaxies. No significant departures of t(cool)/t(ff) below 10 are found. This is inconsistent with models that assume thermally unstable cooling ensues from linear perturbations at or near this threshold. We find that the inner cooling times of cluster atmospheres are resilient to active galactic nucleus (AGN)-driven change, suggesting gentle coupling between radio jets and atmospheric gas. Our analysis is consistent with models in which nonlinear perturbations, perhaps seeded by AGN-driven uplift of partially cooled material, lead to cold gas condensation.