I had the same problem for over half year, I couldn't use the warranty service because it had expired 2 months ago when this issue first started.
The Surface RT goes to sleep and sometimes it doesn't wake up when pressing the power button, pressing any key on the keyboard, touching the mousepad etc... and when touching the windows logo, it vibrates but the screen remain off. In addition, sometimes the surface get hot when I left it in that state for long time, it is also more likely to get like that when the Surface goes to sleep by inactivity than when pressing the power button.
In order to wake it up, I have to do a hard reset, which means pressing the power button for 20 about seconds, and waiting 5 minutes before trying to turn it on. Sometimes it didn't work so I had to hold the power button 20 seconds once the Surface RT is off and then press it several times with a 1 sec interval.
So I looked up for information on internet but most of the affected devices were Surfaces 2 Pro and the solutions were either not functional or applicable to my Surface RT. I refreshed/reset the Surface more than 25 times in the hope that something changed and sometimes it did, twice the wlan adapter didn't work properly, and like half of the time the Surface bricked and blue or yellow lines appeared on the screen, needless to say that I couldn't get rid of the main problem and sometimes it happened just after the reset process finished or when rebooting. I also tried installing all the updates, but it didn't work, I was almost convinced it was a hardware problem rather than a software problem and I never got useful help from Microsoft.
So the last resource was resetting it using a recovery usb from another Surface RT and voila! it turns out that that was the whole problem! I guess the recovery partition on my Surface RT was damaged or corrupted, because the problems began when I reset it for the second time. If you cant get a recovery image from another Surface, you can download it from here:
(Sorry I can't post links (****ty forum), just google "Surface RT startup error 0xc000000d" and select the Microsoft result)
and follow the instructions in order to fix the problem. I chose repartition, and fully recycle the disc in the options of the recovery setup.
I have so far 2 days without any problem.