We have used Raman spectroscopy to study the behavior of double-walled carbon nanotubes (DWNT) under hydrostatic pressure. We find that the rate of change of the tangential mode frequency with pressure is higher for the sample with traces of polymer compared to the pristine sample. We have performed classical molecular dynamics simulations to study the collapse of single (SWNT) and double-walled carbon nanotube bundles under hydrostatic pressure. The collapse pressure (p(c)) was found to vary as 1/R-3, where R is the SWNT radius or the DWNT effective radius. The bundles showed similar to 30% hysteresis and the hexagonally close packed lattice was completely restored on decompression. The p, of a DWNT bundle was found to be close to the sum of its values for the inner and the outer tubes considered separately as SWNT bundles, demonstrating that the inner tube supports the outer tube and that the effective bending stiffness of DWNT, D-DWNT similar to 2D(SWNT).