Most prior work on congestion in datagram systems focuses on buffer management. We find it illuminating to consider the case of a packet switch with infinite storage. Such a packet switch can never run out of buffers. It can, however, still become congested. The meaning of congestion in an infinite-storage system is explored. We demonstrate the unexpected result that a datagram network with infinite storage, first-in, first-out queueing, at least two packet switches, and a finite packet lifetime will, under overload, drop all packets. By attacking the problem of congestion for the infinite-storage case, we discover new solutions applicable to switches with finite storage.