Direct-drive, Rayleigh–Taylor (RT) growth experiments were performed using planar plastic targets on the OMEGA Laser Facility [T. R. Boehly et al., Opt. Commun. 133, 495 (1997)] at laser intensities between ~2 x 10^14 and ~1.5 x 10^15 W/cm^2. The primary purpose of the experiments was to test fundamental physics in hydrocodes at the range of drive intensities relevant to ignition designs. The target acceleration was measured with a streak camera using side-on, x-ray radiography, while RT growth was measured with a framing camera using face-on radiography. In a laser-intensity range from 2 to 5 x 10^14 W/cm^2, the measured RT growth agrees well with two-dimensional simulations, based on a local model of thermal-electron transport. The RT growth at drive intensities above ~1.0 x 10^15 W/cm^2 was strongly stabilized compared to the local model predictions. The experiments demonstrate that standard simulations, based on a local model of electron thermal transport, break down at peak intensities of ignition designs, although they work well at lower intensities. These results also imply that direct-drive ignition targets are significantly more stable than previously calculated using local electron-transport models at peak intensities of ignition designs. The preheating effects by nonlocal electron transport and hot electrons were identified as some of the stabilizing mechanisms.