  Let $G$ be a simple graph that is properly edge coloured with $m$ colours and
let $\M=\{M_1,\ldots, M_m\}$ be the set of $m$ matchings induced by the colours
in $G$. Suppose that $m\le n-n^{c}$, where $c>9/10$, and every matching in $\M$
has size $n$. Then $G$ contains a full rainbow matching, i.e.\ a matching that
contains exactly one edge from $M_i$ for each $1\le i\le m$. This answers an
open problem of Pokrovskiy and gives an affirmative answer to a generalisation
of a special case of a conjecture of Aharoni and Berger.
  Related results are also found for multigraphs with edges of bounded
multiplicity, and for hypergraphs.
  Finally, we provide counterexamples to several conjectures on full rainbow
matchings made by Aharoni and Berger.
