We present classical molecular dynamics simulations demonstrating that single-wall carbon nanotube (SWNT) bundles collapse under hydrostatic pressure. The collapse pressures obtained as a function of nanotube diameter are in excellent quantitative agreement with new data presented here for small diameter (d approximately 0.8 nm) SWNTs, and the majority of previously published results, although there remain some unreconciled contradictions in the literature. The collapse pressure is found to be independent of the nanotube chirality, and a lower limit on the largest SWNT that remains inflated at atmospheric pressure is established (d>4.16 nm).