We use a set of high-resolution N-body simulations of binary galaxy mergers to show that the morphologies of the tidal features that are seen around a large fraction of nearby, massive ellipticals in the field cannot be reproduced by equal-mass dissipationless mergers; rather, they are well explained by the accretion of disk-dominated galaxies. In particular, the arm- and looplike morphologies of the observed tidal debris can only be produced by the kinematically cold material of the disk components of the accreted galaxies. The tidal features that arise from such "cold-accretion" events onto a massive elliptical are visible for significantly longer timescales than the features produced by elliptical-elliptical mergers (about 1-2 Gyr vs. a few hundred million years). Mass ratios of the order of 1:10 between the accreting elliptical and the accreted disk are sufficient to match the brightness of the observed debris. Furthermore, stellar population synthesis models and simple order-of-magnitude calculations indicate that the colors of the tidal features generated in such minor cold-accretion events are relatively red, in agreement with the observations. The minor cold-accretion events that explain the presence, brightness, and structural and color properties of the tidal debris cause only a modest mass and luminosity increase in the accreting massive elliptical. These results, coupled with the relative statistical frequencies of disk- and bulge-dominated galaxies in the field, suggest that massive ellipticals assemble most of their mass well before their tidal debris forms through the accretion of relatively little, kinematically cold material rather than in very recent, dissipationless major mergers.