Raman spectroscopy is employed to study the pressure response of single-(SWCNTs), double- (DWCNTs) and multi-wall (MWCNTs) carbon nanotube materials. In DWCNTs, the outer tubes exhibit a similar pressure response to SWCNTs, where the tube sensitivity to pressure application is diameter dependent (higher for large diameter tubes). On the other hand, the inner tubes in DWCNTs and the innermost tubes in MWCNTs appear to be much less sensitive to pressure application, indicating the pressure screening inside the outer tubes. The pressure screening effect is strongly enhanced in MWCNTs.