We propose a new technique for upper and lower bounding the throughput and blocking probabilities in queueing networks with buffer capacity constraints, i.e., where some buffers in the network have finite capacity. By studying the evolution of multinomials of the state of the system in its steady state, we obtain linear programs whose values upper and lower bound the performance measure of interest, namely throughput or blocking probabilities. The main advantages of this new technique are that the computational complexity does not increase with the size of the finite buffers and that the technique is applicable to systems in which some buffers have infinite capacity. The technique is demonstrated on examples taken from both manufacturing systems and communication networks. As a model for further analysis, for the M/M/s/s queue, we establish that the bounds on the blocking probability are asymptotically tight, i.e., they asymptotically approach the exact value as the degree of the multinomials considered is increased to infinity.