Institutional rules create difficulties for the allocation of moral responsibility. One problem is the existence of responsibility voids, i.e. situations in which an outcome results from individual interactions but for which no one is responsible. Another is that responsibility can be fragmented in the sense that responsibility-bearing individuals may be responsible for different features of the outcome. This study examines both problems together. We show that for a large class of situations the two problems are logically dependent. More precisely, non-dictatorial decision procedures can only ensure the absence of voids if they allow for the fragmentation of responsibility.