I recently experienced the same issue and immediately checked the internet. Soon after I called Apple support. (The best symptom description was from John Stein's video provided on this thread.) They did their remote testing of my iPhone 6s Plus (128GB) and basically offered to make an appointment at the Genius Bar or send me a (new) replacement device. I opted for them to send me a replacement device, which required me to provide a credit card (hold the value of the phone on the card until the old one is received). I have yet to receive the new device, but it seems if Apple can't see any physical damage to the device they're going to replace it via the warranty.