This paper considers a fluid queueing system, fed by N independent sources that alternate between silence and activity periods. We assume that the distribution of the activity periods of one or more sources is a regularly varying function of index ¿. We show that its fat tail gives rise to an even fatter tail of the buffer content distribution, viz., one that is regularly varying of index ¿ + 1. In the special case that ¿ e (-2, -1), which implies long-range dependence of the input process, the buffer content does not even have a finite first moment. As a queueing-theoretic by-product of the analysis of the case of N identical sources, with N ¿ 8, we show that the busy period of an M/G/8 queue is regularly varying of index ¿ iff the service time distribution is regularly varying of index ¿.