We present the results of sensitive observations of the inner 3.3′ (~100 kpc) of 21 clusters of galaxies with the Arecibo telescope utilizing the autocorrelation spectrometer in complex-sampling mode to obtain a total bandwidth of 40 MHz, which corresponds to a velocity coverage of ~7000-8000 km s^-1. These are the most sensitive observations capable of detecting H i with line widths expected for gas in the intracluster medium (ICM). We did not detect any significant emission features with 500 km s^-1 <= FWHM <= 4000 km s^-1 through the centers of these clusters. Limits on optically thin H i from ``narrow'' galaxy-type features are typically ~5 x 10^8 M_☉, while from broad cluster-type features they are about an order of magnitude higher. We derive constraints on the covering factor of any optically thick H i within the Arecibo beam. We obtain upper limits c_f ~ 3 x 10^-3 assuming that the temperature of the H i is 10 K. This corresponds to a limited range of parameter space for the clouds in which the cloud temperature is typically less than a few tens of kelvins and the number of clouds along the line of sight is less than a few tens of clouds. Cloud temperatures this low are probably inconsistent with cloud heating due to X-rays from the ICM; however, this is uncertain. The very small range of parameter space that is allowed by the observations makes it unlikely that there is very much atomic hydrogen in the ICM of these clusters. The Arecibo Observatory is part of the National Astronomy and Ionosphere Center, which is operated by Cornell University under cooperative agreement with the National Science Foundation.