We propose a stubborn design paradigm for nonlinear high-gain observers, resulting in enforcing a saturation nonlinearity with variable limits on the output injection term. We analyze the input-to-state stability properties of the estimation error dynamics, and we show that the stubborn extension does not lead to a worse bound than the original high-gain design in terms of both peaking and sensitivity to disturbances and measurement noise. We illustrate by simulation that the proposed " stubborn " modification is actually effective at improving the transient response of the observer and the rejection of measurement outliers.