  A path in an edge-colored graph is said to be rainbow if no color repeats on
it. An edge-colored graph is said to be rainbow $k$-connected if every pair of
vertices is connected by $k$ internally disjoint rainbow paths. The rainbow
$k$-connection number $\mathrm{rc}_k(G)$ is the minimum number of colors $\ell$
such that there exists a coloring with $\ell$ colors that makes $G$ rainbow
$k$-connected. Let $f(k,t)$ be the minimum integer such that every $t$-partite
graph with part sizes at least $f(k,t)$ has $\mathrm{rc}_k(G) \le 4$ if $t=2$
and $\mathrm{rc}_k(G) \le 3$ if $t \ge 3$. Answering a question of Fujita, Liu
and Magnant, we show that
  \[
  f(k,t) = \left\lceil \frac{2k}{t-1} \right\rceil
  \]
  for all $k\geq 2$, $t\geq 2$. We also give some conditions for which
$\mathrm{rc}_k(G) \le 3$ if $t=2$ and $\mathrm{rc}_k(G) \le 2$ if $t \ge 3$.
