We report high resolution electron microscope (HREM) observations and atomistic simulations of the bending of single and multi-walled carbon nanotubes under mechanical duress. Single and multiple kinks are observed at high bending angles. Their occurrence is quantitatively explained by the simulations, which use a realistic many-body potential for the carbon atoms. We show that the bending is fully reversible up to very large bending angles, despite the occurrence of kinks and highly strained tube regions. This is due to the remarkable flexibility of the hexagonal network, which resists bond breaking and bond switching up to very high strain values.