Suppose that we have an information storage network with m users and n disks, each disk having the same capacity. Different users are connected to arbitrary (perhaps overlapping) subsets of the n disks, and some of the disks might fail. We wish to encode a binary information sequence such that for a specified m-tuple (X/sub 1/, . . . . X/sub m/), the ith user can reliably recover the first X/sub i/ bits of the sequence. There is a natural upper bound on each individual X/sub i/. If this bound can be attained simultaneously for each of the m users, we say that sequential refinement is possible. We find necessary and sufficient conditions for a storage network to admit sequential refinement.