I guess this is quite easy, but I don't see a counterexample:
let $X$ be a noetherian scheme (maybe with more hypotheses, but I don't think this will change much), then I have the feeling that it is not true that every extension of a vector bundle by a vector bundle is again a vector bundle, i.e. if
$0 \rightarrow F \rightarrow G \rightarrow H \rightarrow 0$
is an exact sequence of modules on $X$ with $F$ and $H$ locally free of finite rank, then is does not follow that $G$ is again locally free of finite rank?