  We consider Khintchine type inequalities on the $p$-th moments of vectors of
$N$ $k$-wise independent Rademacher random variables. We show that an analogue
of Khintchine's inequality holds, with a constant $N^{1/2-k/2p}$, when $k$ is
even. We then show that this result is sharp for $k=2$; in particular, a
version of Khintchine's inequality for sequences of pairwise Rademacher
variables \emph{cannot} hold with a constant independent of $N$. We also
characterize the cases of equality and show that, although the vector achieving
equality is not unique, it is unique (up to law) among the smaller class of
exchangable vectors of pairwise independent Rademacher random variables. As a
fortunate consequence of our work, we obtain similar results for $3$-wise
independent vectors.
