Recently, the LHCb Collaboration reported pronounced structures in the invariant mass spectrum of J/ψ 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/ψJ/ψ and ψ(2S)J/ψ, as long as energy-dependent interactions in these channels are allowed, or (ii) with three channels J/ψJ/ψ, ψ(2S)J/ψ, and ψ(3770)J/ψ with just constant contact interactions. Both formulations hint at the existence of a near-threshold state in the J/ψJ/ψ 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 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 η_{c}η_{c} molecule with a similar binding energy.