Shallow cumulus clouds in the trade-wind regions cool the planet by reflecting solar radiation. The response of trade cumulus clouds to climate change is a major uncertainty in climate projections. Trade cumulus feedbacks in climate models are governed by changes in cloud fraction near cloud base, with high climate-sensitivity models suggesting a strong decrease in cloud-base cloudiness due to increased lower-tropospheric mixing. Here we show that novel observations from the EUREC4A field campaign refute this mixing-desiccation hypothesis. We find the dynamical increase of cloudiness through mixing to overwhelm the thermodynamic control through humidity. Because mesoscale motions and the entrainment rate contribute equally to variability in mixing, but have opposing effects on humidity, mixing does not desiccate clouds. The magnitude, variability, and coupling of mixing and cloudiness differ drastically among climate models and with the EUREC4A observations. Models with large trade cumulus feedbacks tend to exaggerate the dependence of cloudiness on relative humidity as opposed to mixing, and also exaggerate variability in cloudiness. Our observational analyses render models with large positive feedbacks implausible, and both support and explain at the process scale a weak trade cumulus feedback. Our findings thus refute an important line of evidence for a high climate sensitivity.