Using the Illustris cosmological simulation, we investigate the origin of fossil groups in the {M}200={10}13{--}{10}13.5 {M}⊙ {h}-1 mass regime. We examine the formation of the two primary features of fossil groups: the large magnitude gap between their two brightest galaxies and their exceptionally luminous brightest group galaxy (BGG). For fossils and nonfossils identified at z = 0, we find no difference in their halo mass assembly histories at early times, departing from previous studies. However, we do find a significant difference in the recent accretion history of fossil and nonfossil halos; in particular, fossil groups show a lack of recent accretion and have in majority assembled 80% of their {M}200(z=0) mass before z˜ 0.4. For fossils, massive satellite galaxies accreted during this period have enough time to merge with the BGG by the present day, producing a more massive central galaxy. In addition, the lack of recent group accretion prevents replenishment of the bright satellite population, allowing for a large magnitude gap to develop within the past few Gyr. We thus find that the origin of the magnitude gap and overmassive BGG of fossils in Illustris depends on the recent accretion history of the groups and merger history of the BGGs after their collapse at z˜ 1. This indicates that selecting galaxy groups by their magnitude gap does not guarantee obtaining either early-forming galaxy systems or undisturbed central galaxies.