The stability and overturning of fully three-dimensional internal gravity wave packets is examined for a rotating, uniformly stratified Boussinesq fluid that is stationary in the absence of waves. We derive through perturbation theory an integral expression for the mean flow induced by upward-propagating fully localized wave packets subject to Coriolis forces. This induced Bretherton flow manifests as a dipolelike recirculation about the wave packet in the horizontal plane. We perform numerical simulations of fully localized wave packets with the predicted Bretherton flow superimposed, for a range of initial amplitudes, wave-packet aspect ratios, and relative vertical wave numbers spanning the hydrostatic and nonhydrostatic regimes. Results are compared with predictions based on linear theory of wave breaking due to overturning, convection, self-acceleration, and shear instability. We find that nonhydrostatic wave packets tend to destabilize due to self-acceleration, eventually overturning although the initial amplitude is well below the overturning amplitude predicted by linear theory. Strongly hydrostatic waves, propagating almost entirely in the horizontal, are found not to attain amplitudes sufficient to become shear unstable, overturning instead due to localized steepening of isopycnals. Results are discussed in the broader context of previous studies of one-and two-dimensional wave packets overturning and recent observations of oceanic internal waves.