Bandwidth-sharing networks as considered by Massoulié and Roberts provide a natural modeling framework for describing the dynamic flow-level interaction among elastic data transfers. Under mild assumptions, it has been established that a wide family of so-called a-fair bandwidth-sharing strategies achieve stability in such networks provided that no individual link is overloaded. In the present paper we focus on a-fair bandwidth-sharing networks where the load on one or several of the links exceeds the capacity. Evidently, a well-engineered network should not experience overload, or even approach overload, in normal operating conditions. Yet, even in an adequately provisioned system with a low nominal load, the actual traffic volume may significantly fluctuate over time and exhibit temporary surges. Furthermore, gaining insight into the overload behavior is crucial in analyzing the performance in terms of long delays or low throughputs as caused by large queue build-ups. The way in which such rare events tend to occur, commonly involves a scenario where the system temporarily behaves as if it experiences overload. In order to characterize the overload behavior, we examine the fluid limit, which emerges from a suitably scaled version of the number of flows of the various classes. The convergence of the scaled number of flows to the fluid limit is empirically validated through simulation experiments. Focusing on linear solutions to the fluid-limit equation, we derive a fixed-point equation for the corresponding asymptotic growth rates. It is proved that a fixed-point solution is also a solution to a related strictly concave optimization problem, and hence exists and is unique. We use the fixed-point equation to investigate the impact of the traffic intensities and the variability of the flow sizes on the asymptotic growth rates. The results are illustrated for linear topologies and star networks as two important special cases. Finally, we briefly discuss extensions to models with user impatience.