[AT&T Samsung Galaxy Note I717] herculeesjr:
Am having this same issue with my girlfriends phone just as of a couple hours ago. It's completely stock and just recently received an OTA update a few days ago, she showed me bragging how she can watch videos and do other things at the same time now knowing how much I want a Note lol, she's had zero issues with it other than the battery getting very hot while charging a week or so ago and seems to be very slightly bulging now that I've looked at it. Out of nowhere today as we're walking through a parking lot and she's carrying it in her hand, not even using it, she says something is wrong with her phone and hands it to me with it vibrating every couple of seconds and not reacting to any buttons. I take the battery out and when we get inside put it back in and without even pressing anything it goes to the Samsung logo, black, animated Samsung logo, then turns off and goes back to vibrating every couple seconds. I look up the button combination for recovery and download mode and all that, I try recovery and it does the same vibrate every couple of seconds but never goes to recovery, and if I take the battery out, hold down volume down, then put the battery in it comes up with the warning about installing ROMs and all that, weather I press volume down or do nothing it stays at that screen for maybe five seconds then turns off and the only way I can get to the download mode is to take the battery out then press volume down. I have the Sprint Samsung Epic 4G and the batteries aren't the same but are very similar, mine is just shorter and 1500mAh, hers is taller and 2500mAh. So for the heck of it I put hers in my phone and hold it enough to where it connects and turn on my phone without issue, it boots up and shows 70% on the battery, I now have it plugged in and charging the battery at the moment with no issues. I put my battery in her phone and held it till it connects and it does the same Samsung logo, goes black, Samsung animated logo, then turns off BUT no vibrating anymore. If I press ANY button though as long as it's being held it vibrates every couple of seconds and if I hold volume down while I connect the battery it goes to Download mode but can't do anything and it turns off after about five seconds same as before, still no recovery access either. Heard that there is a power button issue with this phone and for the heck of it I've wiggled and pressed the power button with no charger or battery connected for around half an hour just to see if I can make it better, but no luck.

Hope to hear a solution to this. I would take the back cover off and clean the power button but I'm not sure if that would void the AT&T insurance, not going to try till I know I can get away with it without them knowing because if I can't fix it it's getting sent off for repairs.

EDIT: And sort of following captemo's suggestion I took a metal box, laid a hoodie over it for a bit of padding, then moderately bounced the phone on its power button side (without letting the power button be pressed/pushed, by letting it hang over the edge of the box) for a minute or so as if thinking the power button was stuck pushed in and I was trying to knock it back out, then put its battery back in and what do you know it turns on fine now. Weirdest way I've ever had to fix a phone, but hey it works now so the girlfriend will be happy.