Gas in protostellar disks provides the raw material for giant planet formation and controls the dynamics of the planetesimal-building dust grains. Accurate gas mass measurements help map the observed properties of planet-forming disks onto the formation environments of known exoplanets. Rare isotopologues of carbon monoxide (CO) have been used as gas mass tracers for disks in the Lupus star-forming region, with an assumed interstellar CO/H2 abundance ratio. Unfortunately, observations of T-Tauri disks show that CO abundance is not interstellar, a finding reproduced by models that show CO abundance decreasing both with distance from the star and as a function of time. Here, we present radiative transfer simulations that assess the accuracy of CO-based disk mass measurements. We find that the combination of CO chemical depletion in the outer disk and optically thick emission from the inner disk leads observers to underestimate gas mass by more than an order of magnitude if they use the standard assumptions of interstellar CO/H2 ratio and optically thin emission. Furthermore, CO abundance changes on million-year timescales, introducing an age/mass degeneracy into observations. To reach a factor of a few accuracy for CO-based disk mass measurements, we suggest that observers and modelers adopt the following strategies: (1) select low-J transitions; (2) observe multiple CO isotopologues and use either intensity ratios or normalized line profiles to diagnose CO chemical depletion; and (3) use spatially resolved observations to measure the CO-abundance distribution.