Kinetic microinstabilities in the solar wind arise when the plasma deviates too far from thermal equilibrium. Previously published work has provided strong evidence that the cyclotron, mirror, and parallel and oblique firehose instabilities limit proton (i.e., ionized hydrogen) temperature anisotropy. However, few studies have thoroughly explored whether a less-abundant ion species can also trigger these instabilities. This study considered the possibility of similar instability-driven limits on a-particle (i.e., fully ionized helium) temperature anisotropy. Linear Vlasov analysis was used to derive the expected threshold conditions for instabilities driven by a-particle temperature anisotropy. Measurements in situ of a-particle temperature anisotropy from the Wind spacecraft's Faraday cups were found to be consistent with the limits imposed by these instability thresholds. This strongly suggests that a-particles, which only constitute ~5% of ions in the solar wind, can drive an instability if their temperature anisotropy becomes sufficiently extreme.