  This work examines superradiance in initially inverted clouds of
\textit{multi-level} atoms. We develop a set of equations that can
approximately calculate the temporal evolution of $N$ coupled atoms. This
allows us to simulate clouds containing hundreds of multi-level atoms while
eschewing the assumption and/or approximation of symmetric dipole-dipole
interactions. This treatment is used to explore the effects that dephasing
caused by elastic dipole-dipole interactions, and competition between multiple
transitions have on superradiance. Both of these mechanisms place strong
parametrical restrictions on a given transition's ability to superradiate.
These results are likely important to recent experiments that probe
superradiance in Rydberg atoms.
