Thin polystyrene (PS) liquid films supported by oxidized silicon (SiOx/Si) substrates may be unstable or metastable, depending on the film thickness, h. In the metastable thickness regime, holes nucleate throughout the surface of the films and subsequently grow under the action of capillary forces. Recent studies show that the hole growth rate in thin PS films is suppressed with the addition of small concentrations Of C-60 fullerenes, due to pinning at the line of contact. We examined the hole growth dynamics in thin film polystyrenes with functionalized single-walled carbon nanotubes (PS-SWNT) supported by SiOx/Si substrates. The hole growth velocities in PS films containing 0.75 wt % functionalized single-walled nanotubes, VPS-SWNT, were appreciably slower than V-PS, the hole-growth velocities of holes in polystyrene films of the same thickness. Moreover, VPS-SWNT and V-PS decreased with decreasing film thickness, for h < 50 nm, with thickness dependencies which exceed theoretical predictions. In addition, V-PS(h)/VPS-SWNT(h) increased with decreasing h, for h < 50 nm, and approached a constant value for larger h. We show that the suppression of the hole-growth rates in the PS-SWNT films are associated with larger viscosities of the PS-SWNT films and that the film thickness -dependent velocities are associated with film thickness -dependent viscosities.