Thermal nonequilibrium (TNE) is a condition of the plasma in the solar corona in which the local rate of energy loss due to radiation increases to the point that it cannot be sustained by the various heating terms acting on the plasma, precluding the existence of a steady state. The limit cycles of precipitation and evaporation that result from TNE have been simulated in 1D models of coronal loops, as well as 2D and 3D models of the solar chromosphere and lower corona. However, a careful study of TNE in the solar wind has not been performed until now. Here, we demonstrate that for suitable combinations of local and global heating rates, it is possible for the plasma to exhibit a TNE condition, even in the context of a transonic solar wind with appreciable mass and energy fluxes. This implies limits on the amount of footpoint heating that can be withstood under steady-state conditions in the solar wind, and may help to explain the variability of solar wind streams that emanate from regions of highly concentrated magnetic flux on the solar surface. The implications of this finding pertain to various sources of high-density solar wind, including plumes that form above regions of mixed magnetic polarity in polar coronal holes and the slow solar wind that emanates from coronal hole boundaries.