Features below the first conductance plateau in ballistic quantum point contacts (QPCs) are often ascribed to electron interaction and spin effects within the single mode limit. In QPCs with a highly asymmetric geometry, we observe sharp resonance peaks when the point contacts are gated to the single mode regime, and surprisingly, under certain gating conditions, a complete destruction of the 2e2/h first quantum plateau. The temperature evolution of the resonances suggest non-Fermi-liquid behavior, while the overall nonlinear characterizations reveal features reminiscent of the 0.7 effect. We attribute these unusual behaviors to the formation of a quasibound state, which is stabilized by a momentum mismatch accentuated by asymmetry. © 2012 American Physical Society.