We show on very general grounds that for a sufficiently high density of states (typically greater than or equal to 100 per cm(-1)) but low enough energy (so that chaos does not play a role) most quantum eigenstates of moderate sized valence-bonded molecules are likely to be delocalized by tunneling interactions between would-be localized states. One implication of this is that energy is essentially never trapped in localized modes or bonds in polyatomic molecules above a threshold energy. Another implication is that spectroscopically prepared zero-order states, which typically possess some local quality, will bear the imprint of the tunneling and spawn narrow spectral clusters whose width is characteristic of the tunneling.