  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.
