  Recently, the LHCb Collaboration reported pronounced structures in the
invariant mass spectrum of $J/\psi$-pairs produced in proton-proton collisions
at the Large Hadron Collider. In this Letter, we argue that the data can be
very well described within two variants of a coupled-channel approach employing
$T$-matrices consistent with unitarity: (i) with just two channels, $J/\psi
J/\psi$ and $\psi(2S)J/\psi$, as long as energy-dependent interactions in these
channels are allowed, or (ii) with three channels $J/\psi J/\psi$,
$\psi(2S)J/\psi$ and $\psi(3770)J/\psi$ with just constant contact
interactions. Both formulations hint at the existence of a near-threshold state
in the $J/\psi J/\psi$ system with the quantum numbers $J^{PC}=0^{++}$ or
$2^{++}$, which we refer to as $X(6200)$. We suggest experimental tests to
check the existence of this state and discuss what additional channels need to
be studied experimentally to allow for distinctive tests between the two
mechanisms proposed. If the molecular nature of the $X(6200)$, as hinted by the
three-channel approach, is confirmed, many other double-quarkonium states
should exist driven by the same binding mechanism. In particular, there should
be an $\eta_c\eta_c$ molecule with a similar binding energy.
