In this paper we study the cell loss asymptotics for finite buffers accessed by a large number of stationary independent sources and which are served according to a strict HOL priority rule. We first consider the case of two buffers with one of them having strict priority over the other and we obtain asymptotically exact expressions for the cell loss probability for the queues. The asymptotics are studied in terms of a scaling parameter which reflects the server speed, buffer size and the number of sources in such a way that the ratios remain constant. Moreover, as in the single queue case the results are valid for long-range dependent sources with bounded instantaneous rates. The results are then generalised to the case of M buffers where it is shown that resource pooling takes place by which all higher order priority queues can be lumped together when calculating the asymptotics of the lowest priority queue. We conclude with some numerical validation of our formulae against simulations which confirm the theory.