In this paper, we present the results of a VLA search for H I absorption with high spectral resolution (1.6 km s-1) towards A496, A1795, and A2584 (and A2597 taken from O'Dea, Baum, & Gallimore). These observations are well matched to the properties of cold, optically thick H I clouds, where the line width is given by the width of an individual cloud rather than the dispersion in an ensemble of clouds. We do not detect any H I absorption with narrow linewidths in these clusters. Our limits mainly apply to clouds which are larger than a few tenths of a parsec - i.e., if the clouds are much smaller than the background radio source and have a low covering factor in velocity space, they could still escape detection. The estimated limits on column density (for clouds in this regime of parameter space) are 2-3 orders of magnitude less than the 1021 cm-2 required to explain the x-ray absorption seen in some cooling flow clusters. The combination of our high spectral resolution H I absorption searches with the existing lower spectral resolution H I absorption searches and the searches for H I emission makes it unlikely that atomic hydrogen is the dominant component of the cold x-ray absorbing gas in the ICM.