I'm having the same problem. I know it's got plenty of juice, and if I try to use an app (some apps, not all), it will turn off. When I turn it back on, it bleeps at me and says the battery is critically low and needs to be recharged, then usually turns off again. I plug it in; at first the battery image will come up empty, then a couple of minutes later it says 60% or whatever it has. I can then turn it back on and it works. But certain apps set it off: playing Tetris (paid, not free, so not loading ads); Nook; and Google Play Magazines. I only found the last two today so there might be more. But I've been using Podkicker Pro daily on the train on the way to work--obviously it's not plugged in--and it's fine. ETA: The podcasts are downloaded, not streaming.
Since all of these apps are full-screen, I can't tell if I lose the signal first, and to be honest my Sprint signal, phone and 4G, sucks most of the time so I probably wouldn't notice the difference.
I'm running stock, no root or ROM, 4.0.4, FH 13
ETA the second: can a mod please retitle this post so it says the problem, in case others are having it?