We analyze the potential of the Fermilab Tevatron and CERN Large Hadron Collider (LHC) to study anomalous quartic vector--boson interactions (photon photon Z Z) and (photon photon W+ W-). Working in the framework of SU(2)_L X U(1)_Y chiral Lagrangians, we study the production of photons pairs accompanied by (e+e-), (e nu), and jet pairs to impose bounds on these new couplings, taking into account the unitarity constraints. We compare our findings with the indirect limits coming from precision electroweak measurements as well as with presently available direct searches at LEPII. We show that the Tevatron Run II can provide limits on these quartic limits which are of the same order of magnitude as the existing bounds from LEPII searches. LHC will be able to tighten considerably the direct constraints on these possible new interactions, leading to more stringent limits than the presently available indirect ones.