We present an analysis of 55 central galaxies in clusters and groups with molecular gas masses and star formation rates lying between $10^{8}-10^{11} M_{odot}$ and $0.5-270$ $M_{odot} yr^{-1}$, respectively. We have used Chandra observations to derive profiles of total mass and various thermodynamic variables. Molecular gas is detected only when the central cooling time or entropy index of the hot atmosphere falls below $sim$1 Gyr or $sim$35 keV cm$^2$, respectively, at a (resolved) radius of 10 kpc. This indicates that the molecular gas condensed from hot atmospheres surrounding the central galaxies. The depletion timescale of molecular gas due to star formation approaches 1 Gyr in most systems. Yet ALMA images of roughly a half dozen systems drawn from this sample suggest the molecular gas formed recently. We explore the origins of thermally unstable cooling by evaluating whether molecular gas becomes prevalent when the minimum of the cooling to free-fall time ratio ($t_{rm cool}/t_{rm ff}$) falls below $sim10$. We find: 1) molecular gas-rich systems instead lie between $10 < min(t_{rm cool}/t_{rm ff}) < 25$, where $t_{rm cool}/t_{rm ff}=25$ corresponds approximately to cooling time and entropy thresholds $t_{rm cool} lesssim 1$ Gyr and 35 keV~cm$^2$, respectively, 2) $min(t_{rm cool}/t_{rm ff}$) is uncorrelated with molecular gas mass and jet power, and 3) the narrow range $10 < min(t_{rm cool}/t_{rm ff}) < 25$ can be explained by an observational selection effect. These results and the absence of isentropic cores in cluster atmospheres are in tension with precipitation models, particularly those that assume thermal instability ensues from linear density perturbations in hot atmospheres. Some and possibly all of the molecular gas may instead have condensed from atmospheric gas lifted outward either by buoyantly-rising X-ray bubbles or merger-induced gas motions.
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