Possibilistic logic is a well-known framework for dealing with uncertainty and reasoning under inconsistent or prioritized knowledge bases. This paper deals with conditioning uncertain information where the weights associated with formulas are in the form of sets of uncertainty degrees. The first part of the paper studies set-valued possibility theory where we provide a characterization of set-valued possibilistic logic bases and set-valued possibility distributions by means of the concepts of compatible possibilistic logic bases and compatible possibility distributions respectively. The second part of the paper addresses conditioning set-valued possibility distributions. We first propose a set of three natural postulates for conditioning set-valued possibility distributions. We then show that any set-valued conditioning satisfying these three postulates is necessarily based on conditioning the set of compatible standard possibility distributions. The last part of the paper shows how one can efficiently compute set-valued conditioning over possibilistic knowledge bases.