We consider the possibility that the Magellanic Clouds were the largest members of a group of dwarf galaxies that entered the Milky Way (MW) halo at late times. Seven of the eleven brightest satellites of the MW may have been part of this system. The proximity of some dwarfs to the plane of the orbit of the Large Magellanic Cloud (LMC) has been used to argue that they formed from tidal debris from the LMC and Small Magellanic Cloud (SMC). Instead, they may owe to the tidal breakup of the Magellanic group. This can explain the association of many of the dwarf galaxies in the Local Group with the LMC system. It provides a mechanism for lighting up dwarf galaxies and reproduces the bright end of the cumulative circular velocity distribution of the satellites in the MW without invoking a stripping scenario for the subhalos to match the satellite distribution expected according to CDM theory. Finally, our model predicts that other isolated dwarfs will be found to have companions. Evidence for this prediction is provided by nearby, recently discovered dwarf associations.