Multiple-weak-boson signals expected at future hadron supercolliders from several sources will provide new ways to probe the underlying interactions. The production of {ital t{bar t}t{bar t}}, {ital t{bar t}} {ital H}{sup 0}, {ital H}{sup 0}{ital H0}, {ital t{bar t}} {ital W}{sup +}{ital W{minus}}, {ital t{bar t}Z}{sup 0}{ital Z0}, Z{sup 0}{ital Z0}, {ital t{bar t}} {ital Z}{sup 0}, and {ital t{bar t}} {ital W}{sup {plus minus}} by standard-model mechanisms will lead to distinctive three-boson and four-boson signals via the decays {ital t}{r arrow}{ital bW} and {ital H}{sup 0}{r arrow}{ital W}{sup +}{ital W{minus}},{ital Z}{sup 0}{ital Z0},{ital t{bar t}},{gamma}{gamma}. In supersymmetric models the production and cascade decay of heavy gluinos can also give multiple {ital W} and {ital Z} configurations. We present calculations of these principal top-quark and Higgs-boson decay channels (plus some less important channels) and compare the resulting multi-weak-boson signals with backgrounds from direct multiboson production at energies of the Superconducting Super Collider and CERN Large Hadron Collider. We also present calculations of typical gluino cascade contributions. Since weak bosons are most readily identified through their leptonic decays, we also evaluate the principal signals and backgrounds at the lepton level. Within the standard model, the signals associated with strong top-quark productionmore » dominate completely over electroweak contributions; the expected event rates are large if we do not insist on all-leptonic {ital W} and {ital Z} decays. In the minimum supersymmetric standard model, gluino cascade decays can lead to still higher event rates; top-quark channels are typically important contributors here too.« less