Using the science verification data of the Dark Energy Survey for a new sample of 106 X-ray selected clusters and groups, we study the stellar mass growth of bright central galaxies (BCGs) since redshift z ∼ 1.2. Compared with the expectation in a semi-analytical model applied to the Millennium Simulation, the observed BCGs become under-massive/under-luminous with decreasing redshift. We incorporate the uncertainties associated with cluster mass, redshift, and BCG stellar mass measurements into an analysis of a redshift-dependent BCG-cluster mass relation, , and compare the observed relation to the model prediction. We estimate the average growth rate since z = 1.0 for BCGs hosted by clusters of M_200,z = 10^13.8 M_o; at z = 1.0: m_∗,BCG appears to have grown by 0.13 ±0.11 dex, in tension at the ∼2.5σ significance level with the 0.40 dex growth rate expected from the semi-analytic model. We show that the build-up of extended intracluster light after z = 1.0 may alleviate this tension in BCG growth rates.