We present the results of a search for neutral hydrogen in 14 cooling flow and two non-cooling flow clusters of galaxies with the Arecibo 1000 foot and Green Bank 300 foot radio telescopes. H I was detected in absorption against the radio continuum sources of two (2A0335 + 096 and MKW3s) centrally dominant, cooling flow cluster galaxies (CFDs). The absorption features have column densities of NH I, = 2-3 ×1020 (Ts/100 K)-1 cm-2, widths of ΔVFWHM ∼ 30-100 km s-1, and optical depths of a few percent. The H I absorption features are redshifted with respect to the CFDs by ∼100-250 km s-1, indicating that the H I is probably falling into the CFDs. The H I in MKW3s appears to be located at a projected distance of ∼70 kpc from the nucleus of the CFD. H I was not detected in emission; the mean H I upper limit for the sample is MH I, ≲ 6 × 109 M⊙ of optically thin H I, but the limit based on the most sensitive observations is MH I ≲ 2 × 108 M⊙. Upper limits to the H I to dust ratio in four CFDs are consistent with the Galactic mean gas-to-dust ratio (Mg/Md ∼100). The emission limits suggest that the optically thin H I is confined to 8-15 kpc regions about the radio sources. If the H I is in pressure equilibrium with the high-temperature and high-pressure X-ray plasma, then the H I clouds are orders of magnitude smaller, denser, and less massive than their Galactic counterparts. The small amount of H I and the large star formation rate observed in 2A 0335 + 096 suggest that star formation occurs more rapidly and efficiently than in disk galaxies. The origin of the H I cannot be determined with certainty, but the large cooling rates in the detected CFDs (〈ṁCF〉det = 250 M⊙ yr-1) compared to the mean for the CFDs that were not detected (〈ṁCF〉nd = 106 M⊙ yr-1) suggests that the H I originated from the cooling flows. These observations, together with the recent detections of H I and CO in NGC 1275, constitute the most direct evidence to date that the intracluster gas is accreting onto the CFDs and cooling to temperatures low enough for star formation to occur.