Dispersion of carbon nanotubes (CNTs) into liquids typically requires ultrasonication to exfoliate individuals CNTs from bundles. Experiments show that CNT length drops with sonication time (or energy) as a power law t(-m). Yet the breakage mechanism is not well understood, and the experimentally reported power law exponent m ranges from approximately 0.2 to 0.5. Here we simulate the motion of CNTs around cavitating bubbles by coupling brownian dynamics with the Rayleigh-Plesset equation. We observe that, during bubble growth, CNTs align tangentially to the bubble surface. Surprisingly, we find two dynamical regimes during the collapse: shorter CNTs align radially, longer ones buckle. We compute the phase diagram for CNT collapse dynamics as a function of CNT length, stiffness, and initial distance from the bubble nuclei and determine the transition from aligning to buckling. We conclude that, depending on their length, CNTs can break due to either buckling or stretching. These two mechanisms yield different power laws for the length decay (0.25 and 0.5, respectively), reconciling the apparent discrepancy in the experimental data.