Many measurements in cosmology depend on the use of integrated distances or time, but. galaxies evolving passively on a timescale much longer than their age difference allow us to determine the expansion rate H(z) solely as a function of the redshift-time derivative dz/dt. These model-independent "cosmic chronometers" can therefore be powerful discriminators for testing different cosmologies. In previous applications, the available sources strongly disfavored models (such as Lambda CDM) predicting a variable acceleration, preferring instead a steady expansion rate over the redshift range 0 less than or similar to z less than or similar to 2. A more recent catalog of 30 objects appears to suggest non-steady expansion. In this paper, we show that such a result is entirely due to the inclusion of a high, locally inferred value of the Hubble constant H-0 as an additional datum in a set of otherwise pure cosmic-chronometer measurements. This H-0, however, is not the same as the background Hubble constant if the local expansion rate is influenced by a Hubble Bubble. Used on their own, the cosmic chronometers completely reverse this conclusion, favoring instead a constant expansion rate out to z similar to 2.