The whistler anisotropy instability is driven by the condition T$perp$e/T$parallel$e > 1, where the subscript e denotes electrons and the other subscripts denote directions relative to the background magnetic field Bo. Instability growth leads to enhanced field fluctuations which scatter the electrons; theory and simulations show that this scattering imposes an upper bound on the electron anisotropy in the form T$perp$e/T$parallel$e - 1 = Se/¿$parallel$eαe with fitting parameters 0.1 $lesssim$ Se $lesssim$ 1 and 0.5 $lesssim$ αe < 0.7 over 0.10 ≤ ¿$parallel$e ≤ 1.0 where ¿$parallel$e = 8πneT$parallel$e/Bo2. Here measurements from the PEACE instrument on the Cluster 1 spacecraft show that electron anisotropies in two crossings of the dayside terrestrial magnetosheath are constrained statistically by this equation with Se $simeq$ 0.2 and αe $simeq$ 0.6. This is the first reported observation of this constraint in a space plasma.