Using empirical potentials and atomistic simulations, we model three-point bend tests of single-walled carbon nanotubes encapsulating metal nanowires. The presence of a metal nanowire inside the nanotube greatly suppresses the tube-buckling instability. Increasing tube diameter leads to an increase in the bending strength; however, in contrast to hollow tubes, there is no decrease in the maximum deflection before buckling. Analysis of the principal bending vibrational mode shows a lowering of the frequency, associated with increased tube inertia. Remarkably, metal-filled tubes exhibit strong damping of oscillations whereas unfilled single-walled and multiwalled tubes show no damping. Our studies demonstrate the benefits of filling tubes with solids to modify bending strength and flexibility, suggesting applications for nanotube-based elements in micromechanical devices or nanoprobes.