A recent series of experiments on the OMEGA laser provided the first controlled demonstration of the KelvinHelmholtz instability in a high-energy-density physics context (E. C. Harding, et al., Phys. Rev. Lett., 103, 045005,2009; O. A. Hurricane, et al., Phys. Plasmas, 16, 056305, 2009). We present 3D simulations which resolve previously reported discrepancies between those experiments and the 2D simulation used to design them. Our new simulations reveal a three-dimensional mechanism behind the low density "bubble" structures which appeared in the experimental x-ray radiographs at late times but were completely absent in the 2D simulations. We also demonstrate that the three-dimensional expansion of the walls of the target is sufficient to explain the 20% overprediction by 2D simulation of the late-time growth of the KH rollups. The implications of these results for the design of future experiments is discussed.