  Using the Illustris cosmological simulation, we investigate the origin of
fossil groups in the $M_{200}=10^{13}-10^{13.5}M_{\odot}/h$ 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 non-fossils identified
at $z=0$, we find no difference in their halo mass assembly at early times,
departing from previous studies. However, we do find a significant difference
in the recent accretion history of fossil and non-fossil 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 $z\sim 0.4$. 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; and, 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 BGG after their collapse at $z\sim1$. This indicates that
selecting galaxy groups by their magnitude gap does not guarantee obtaining
neither early-forming galaxy systems nor undisturbed central galaxies.
