The stability and overturning of fully three-dimensional internal gravity wavepackets 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 wavepackets subject to Coriolis forces. This induced "Bretherton flow" manifests as a dipole-like recirculation about the wavepacket in the horizontal plane. We perform numerical simulations of fully localized wavepackets with the predicted Bretherton flow superim- posed, for a range of initial amplitudes, wavepacket aspect ratios, and relative vertical wavenum- bers spanning the hydrostatic and non-hydrostatic 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 non-hydrostatic wavepackets tend to de-stabilize due to self-acceleration, eventually overturning although the initial amplitude is well below the overturning amplitude pre- dicted 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 wavepacket overturning, and recent observations of oceanic internal waves.