  Let $G$ be a group such that $G/Z(G)$ is finite and simple. The
non-commuting, non-generating graph $\Xi(G)$ of $G$ has vertex set $G \setminus
Z(G)$, with edges corresponding to pairs of elements that do not commute and do
not generate $G$. We show that $\Xi(G)$ is connected with diameter at most $5$,
with smaller upper bounds for certain families of groups. When $G$ itself is
simple, we prove that the diameter of the complement of the generating graph of
$G$ has a tight upper bound of $4$. In the companion paper arXiv:2211.08869, we
consider $\Xi(G)$ when $G/Z(G)$ is not simple.
