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Drisko (1998) proved (essentially) that every family of 2 n − 1 matchings of size n in a bipartite graph possesses a partial rainbow matching of size n . In Barát et al. (2017) this was generalized as follows: Any ⌊ k + 2 k + 1 n ⌋ − ( k + 1 ) matchings of size n in a bipartite graph have a rainbow matching of size n − k . The paper has a twofold aim: (i) to extend these results to matchings of not necessarily equal cardinalities, and (ii) to prove a conjecture of Drisko, on the characterization of those families of 2 n − 2 matchings of size n in a bipartite graph that do not possess a rainbow matching of size n . Combining the latter with an idea of Alon (2011), we re-prove a characterization of the extreme case in a well-known theorem of Erdős–Ginzburg–Ziv in additive number theory. [ABSTRACT FROM AUTHOR]