The spillover mechanism of molecular hydrogen on carbon nanotubes in the presence of catalytically active platinum clusters was critically and systematically investigated by using density-functional theory. Our simulation model includes a Pt-4 cluster for the catalyst nanoparticle and curved and planar circumcoronene for two exemplary single-walled carbon nanotubes (CNT), the (10,10) CNT and one of large diameter, respectively. Our results show that the H-2 molecule dissociates spontaneously on the Pt-4 cluster. However, the dissociated H atoms have to overcome a barrier of more than 2 eV to migrate from the catalyst to the CNT, even if the Pt-4 cluster is at full saturation with six adsorbed and dissociated hydrogen molecules. Previous investigations have shown that the mobility of hydrogen atoms on the CNT surface is hindered by a barrier. We find that instead the Pt-4 catalyst may move along the outer surface of the CNT with activation energy of only 0.16 eV, and that this effect offers the possibility of full hydrogenation of the CNT. Thus, although we have not found a low-energy pathway to spillover onto the CNT, we suggest, based on our calculations and calculated data reported in the literature, that in the hydrogen-spillover process the observed saturation of the CNT at hydrogen background pressure occurs through mobile Pt nanoclusters, which move on the substrate more easily than the substrate-chemisorbed hydrogens, and deposit or reattach hydrogens in the process. Initial hydrogenation of the carbon substrate, however, is thermodynamically unfavoured, suggesting that defects should play a significant role.