DIAGNOSIS The proximity sensor (the little black dot near the top) that wakes the screen is either dirty, blocked, or its sensitivity setting has been changed. It's not necessarily broken.
FIX First, give the entire front face, especially the top bezel where the sensor is, a good cleaning as described in the cleaning question. A layer of dust or grime can diffuse the sensor's beam. If that doesn't help, check the settings. 1) Press the ring to bring up the Quick View menu. 2) Go to Settings > Display. 3) Look for "Farsight" or "Screen Brightness" settings. Farsight, if you have it on, keeps the screen brighter longer. More importantly, make sure "Auto-Brightness" is on so it adjusts to your room's light.
IF THAT DIDN'T WORK The sensor itself might be faulty. You can test this in a dark room. The screen should not wake with a wave if it's truly dark—it uses a combination of proximity and ambient light. If it doesn't wake even in a well-lit room after a cleaning and settings check, it may be a hardware issue.
HEADS UP The Nest E is designed to have a fairly dim, sleepy display to blend into your wall and save power. It's not as instantly bright as a phone screen. Also, if the thermostat's internal battery is low (check in Settings > Technical Info > Power), it will make the screen extra conservative about turning on to conserve energy.