We consider a fluid queue fed by the superposition of n homogeneous on-off sources with generally distributed on and off periods. The buffer space B and link rate C are scaled by n, so that we get nb and nc, respectively. Then we let n grow large. In this regime, the overflow probability decays exponentially in the number of sources n. We specifically examine the scenario where b is also large. We obtain explicit asymptotics for the case where the on periods have a subexponential distribution, e.g., Pareto, Lognormal, or Weibull. The results show a sharp dichotomy in the qualitative behavior, depending on the shape of the function v(t) := - logP(A* > t) for large t, A* representing the residual on period. If v(.) is regularly varying of index 0 (e.g., Pareto, Lognormal), then, during the path to overflow, the input rate will only slightly exceed the link rate. Consequently, the buffer will fill ‘slowly’, and the typical time to overflow will be ‘more than linear’ in the buffer size. In contrast, if v(.) is regularly varying of index strictly between 0 and 1 (e.g., Weibull), then the input rate will significantly exceed the link rate, and the time to overflow is roughly proportional to the buffer size. In both cases there is a substantial fraction of the sources that remain in the on state during the entire path to overflow, while the others contribute at their mean rates. These observations lead to approximations for the overflow probability. The approximations may be extended to the case of heterogeneous sources. The results provide further insight into the so-called reduced-load approximation.