The mass of a protoplanetary disk limits the formation and future growth of any planet. Masses of protoplanetary disks are usually calculated from measurements of the dust continuum emission by assuming an interstellar gas-to-dust ratio. To investigate the utility of CO as an alternate probe of disk mass, we use ALMA to survey ¹³CO and C¹⁸O J = 3–2 line emission from a sample of 93 protoplanetary disks around stars and brown dwarfs with masses from in the nearby Chamaeleon I star-forming region. We detect ¹³CO emission from 17 sources and C¹⁸O from only one source. Gas masses for disks are then estimated by comparing the CO line luminosities to results from published disk models that include CO freeze-out and isotope-selective photodissociation. Under the assumption of a typical interstellar medium CO-to-H₂ ratio of 10−⁴, the resulting gas masses are implausibly low, with an average gas mass of ~0.05 M Jup as inferred from the average flux of stacked ¹³CO lines. The low gas masses and gas-to-dust ratios for Cha I disks are both consistent with similar results from disks in the Lupus star-forming region. The faint CO line emission may instead be explained if disks have much higher gas masses, but freeze-out of CO or complex C-bearing molecules is underestimated in disk models. The conversion of CO flux to CO gas mass also suffers from uncertainties in disk structures, which could affect gas temperatures. CO emission lines will only be a good tracer of the disk mass when models for C and CO depletion are confirmed to be accurate.