I have this problem multiple times during each day. Sometimes my iPhone 6+ responds to a hard reboot, and sometimes it does not. Yesterday after multiple attempts to make the screen light up by doing hard reboots, it just sat on the table and repeatedly did the double-buzz that indicates power connection. It still would not turn on. I plugged the phone into the USB on my iMac and iTunes could not find it. Then it found it. I did the online diagnostic and resynced. It worked the rest of the day with the new iOS update installed. Today, it refuses to light up. So I press the wake-up/enter button, then the power button, then the wake-up button, then the power button, and finally I get an unresponsive screen that looks blurry and out of focus. Then I received a phone call, but couldn't answer it because the screen was unresponsive.
In short, a hard reboot does not solve this problem for me. I'm sending the phone off to Apple for repair and expect to be without a phone for the next two weeks. I have not had a problem with an Apple product since I began using them in 1983, so I guess there's a first time for everything.