  Double $J/\psi$ production on the order of 20-30 pb has been observed by the
NA3 collaboration. These $\psi \psi$ pairs, measured in $\pi^- N$ interactions
at 150 and 280 GeV$/c$ and in $pN$ interactions at 400 GeV$/c$, carry a large
fraction of the projectile momentum, $x_{\psi \psi} \geq 0.6$ for the 150
GeV$/c$ beam and $\geq 0.4$ at 280 GeV$/c$. We examine several sources of $\psi
\psi$ pair production within QCD, including ${\cal O}(\alpha_s^4)$ $\psi \psi$
production, leading-twist $b \overline b$ production and decay, and the
materialization of heavy-quark Fock states in the projectile. We estimate the
production cross section and the single and double $J/\psi$ momentum and mass
distributions for each, comparing the results with the NA3 data, and predict
$\psi \psi$ production in $pN$ interactions at 800 GeV$/c$, accessible to
current fixed-target experiments. We also discuss the observable implications
of open heavy meson pair production from the intrinsic heavy quark Fock states.
