Recent X-ray spectroscopic results suggest that 2-3 solar masses/yr of cooling gas may be falling into M87 and as much as 300 solar masses/yr into giant galaxies in rich clusters. The eventual fate of the gas in the radiatively accreting flow is an open question. In an attempt to set some bounds on the evolution of the gas, a sensitive search was performed for H I in a sample of eight D and cD galaxies in rich and poor clusters. Null detections in all cases set upper limits of (4-20) x 10 to the 8th solar masses of H I in the giant galaxies. Possible reasons for these nondetections and alternative evolutionary fates for the accreting gas are discussed.