This work focuses on the high pressure Raman study of carbon nanostuctures comprising of single- or double-wall carbon nanotubes. The detailed examination of the Raman peaks, especially those attributed to the radial breathing modes of the carbon nanotubes, as a function of pressure provides a wealth of information concerning the pressure response of individual nanotubes as well as the internal-external tube (intratube) interactions. The radial breathing modes of both the internal and the external tubes in double-wall carbon nanotubes show reduced pressure slope values as compared to the corresponding in single-wall carbon nanotubes. The reduced slopes for the internal tubes reflect the pressure screening effect inside the external tubes, while those for the external tubes their structural reinforcement due to the encapsulation of smaller diameter tubes in their interior. Moreover, the magnitude of the pressure screening effect depends strongly on the intratube spacing and thus on the intratube interaction. All the experimental observations are well reproduced qualitatively by means of theoretical calculations based on a simple phenomenological model.