Bisemilattice-valued fuzzy set ($B$-fuzzy set) is a mapping from a nonempty set to a bisemilattice. A $B$-fuzzy set has two families of level subsets, one for each ordering of the bisemilattice. In this paper, necessary and sufficient conditions under which two families of subsets of a nonempty set are families of level subsets of a $B$-fuzzy set are given.