For each original data segment, a distributed storage processing unit generates encoded slices designed to prevent the original data segment from being reconstructed using fewer than a threshold number of encoded slices. Multiple encoded slices are generated for each of two different data segments, and the slices associated with the first and second data segment are stored substantially concurrently in different storage sets employing different distributed storage units. Encoded slices for even and odd data segments can be stored in different storage sets, or longer sequences of data segments can be stored in alternating storage sets. Storage sets can also be determined by the vault generation of a particular data segment.