Solar wind electrons are often observed to consist of two distinguishable components, a thermal, more dense core and a suprathermal, less dense halo. In this core/halo model linear Vlasov theory for the whistler heat flux instability predicts dimensionless heat flux thresholds which decrease as the electron core beta, &bgr;˜∥c, increases. It has been proposed that this theoretical threshold corresponds to an observable upper bound on the electron heat flux. Linear theory also predicts that there is a critical value of &bgr;˜∥c below which the whistler heat flux instability does not have appreciable growth in the solar wind; there is another suggestion that this corresponds to an observable lower bound on &bgr;˜∥c. These two proposals are examined by comparison of linear theory and data from the initial in-ecliptic phase of the Ulysses mission. The instability threshold does provide a statistical constraint on observed solar wind heat fluxes, and the critical &bgr;˜∥c of theory is not inconsistent with a statistical lower bound on the observations of that parameter. ¿ 1999 American Geophysical Union