Our central Galactic supermassive black hole, Sgr A*, exists mostly in a very stable, extremely low-luminosity ( 10-9 L_Edd), thermal quiescent state, which is interrupted roughly daily by a brief, nonthermal X-ray flare. Because they are not accompanied by significant changes in the radio wavelengths, the flares make Sgr A* unusual in the context of black holes accreting at slightly higher rates. Those sources display a radio/X-ray luminosity correlation whose normalization scales with central mass, and that holds over orders of magnitude in accretion power. There is significant scatter in this correlation, due in part to measurement uncertainties and intrinsic variability. By studying the correlation in sources bracketing Sgr A* in radio luminosity and whose physical parameters are well measured, we can derive a statistical measure of this local scatter. We find that Sgr A* in quiescence and the lower intensity flares fall well below the correlation in X-ray luminosity. The brightest flares are consistent within the scatter, which may indicate an upper bound on the X-ray luminosity. This trend is suggestive of a state transition at the extreme low end of accretion activity, only above which the radio/X-ray correlation is tracked. This scenario is easily testable because it must fulfill three unique observational predictions: 1) As long as Sgr A* remains at its current radio luminosity, no X-ray flare will be seen which statistically exceeds the prediction of the correlation, 2) no source already on the correlation will be seen to flare in the X-rays similar to Sgr A* (i.e., without corresponding increases in the radio luminosity), and 3) sources below a critical accretion rate or luminosity will show similar flares as Sgr A*, on timescales appropriate to their masses.