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We study the pressure induced collapse of single-, double- and triple-wall carbon nanotubes. Theoretical simulations were performed using density-functional tight-binding theory. For tube walls separated by the graphitic distance, we show that the radial collapse pressure, Pc, is mainly determined by the diameter of the innermost tube, din and that its value significantly deviates from the usual Lévy-Carrier law. A modified expression, with α and β numerical parameters, which reduces the collapse pressure for low diameters is proposed. For din ≳ 1.5 nm an enhanced stability is found which may be assigned as due to the bundle intertube geometry-induced interactions. If the inner and outer tubes are separated by larger distances, the collapse process is found to be more complex. High-pressure resonant Raman experiments were performed in double-wall carbon nanotubes having inner and outer diameters averaging 1.5 nm and 2.0 nm, respectively. A modification in the response of the G-band and the disappearance of the radial breathing modes between 2 GPa and 5 GPa indicate the beginning and the end of the radial collapse process. Experimental results are in good agreement with our theoretical predictions, but do not allow to discriminate from those corresponding to a continuum mechanics model.