In this paper, we consider the question of the existence of disjoint maximal independent sets (MIS) in graphs and hypergraphs. The question was raised in the 1970's independently by C. Berge and C. Payan. They considered the question of characterizing the graphs that admit disjoint MIS, and in particular whether regular graphs do. In this paper, we are interested in the existence of disjoint MIS in a graph or in its complement, motivated by the fact that most constructions of graphs that do not admit disjoint MIS are such that their complement does. We prove that there are disjoint MIS in a graph or its complement whenever the graph has diameter at least three or has chromatic number at most four. We also define a graph of chromatic number 5 and diameter 2 which does not admit disjoint MIS nor its complement. As our work was first motivated by a more recent work on disjoint MIS in hypergraphs by Acharya (2010), we also consider the question of the existence of disjoint MIS in hypergraphs. We answer a question by C. Jose and Z. Tuza (2009), proving that there exists balanced k-connected hypergraphs admitting no disjoint MIS.