Star formation in the universe's largest galaxies---the ones at the centers of galaxy clusters---depends critically on the thermodynamic state of their hot gaseous atmospheres. Central galaxies with low-entropy, high-density atmospheres frequently contain multiphase star-forming gas, while those with high-entropy, low-density atmospheres never do. The dividing line between these two populations in central entropy, and therefore central cooling time, is amazingly sharp. Two hypotheses have been proposed to explain the dichotomy. One points out that thermal conduction can prevent radiative cooling of cluster cores above the dividing line. The other holds that cores below the dividing line are subject to thermal instability that fuels the central AGN through a cold-feedback mechanism. Here we explore those hypotheses with an analysis of the H-alpha properties of ACCEPT galaxy clusters. We find that the two hypotheses are likely to be complementary. Our results support a picture in which cold clouds inevitably precipitate out of cluster cores in which cooling outcompetes thermal conduction and rain down on the central black hole, causing AGN feedback that stabilizes the cluster core. In particular, the observed distribution of the cooling-time to freefall-time ratio is nearly identical to that seen in simulations of this cold-feedback process, implying that cold-phase accretion, and not Bondi-like accretion of hot-phase gas, is responsible for the AGN feedback that regulates star formation in large galaxies. ; Comment: Submitted to ApJ Letters