We show that there exists a bipartite graph containing n matchings of sizes m i n satisfying i m i = n 2 + n/2 − 1, such that the matchings have no rainbow matching. This answers a question posed by Aharoni, Charbit and Howard. We also exhibit (n − 1) × n latin rectangles that cannot be decomposed into transversals, and some related constructions. In the process we answer a question posed by Häggkvist and Johansson. Finally, we propose a Hall-type condition for the existence of a rainbow matching.