Changes in anvil cloud area with warming are a leading source of uncertainty in estimating the Earth's climate sensitivity (Forster et al 2021). Most approaches to bounding this area feedback rely on climate models or expert assessment. Here, we use observations and theory, a "storyline approach", to bound it. We first derive a simple but quantitative expression for the anvil area feedback, which is shown to depend on the present day, measurable cloud radiative effects and the fractional change in anvil area with warming. Satellite observations suggest an anvil cloud radiative effect of about \(\pm\) 1 Wm^-2, which requires the fractional change in anvil area to be about \(\mp\) 50 % K^-1 to produce a feedback equal to its present-day lower bound. We use theory and observations to show that the change in anvil area is closer to about \(-\)4 % K^-1. This rules out the previous estimate of the area feedback and leads to our new estimate of 0.02 \(\pm\) 0.07 Wm^-2K^-1 which is many times weaker and more constrained. In comparison, we show the anvil cloudy albedo feedback to be much less constrained. This poses an obstacle for bounding the Earth's climate sensitivity.