This has happened to my phone twice - the first time Samsung replaced the LCD and shipped it back. A week later the same thing happened again:
Phone was working fine. Battery drained completely. After plugging in to charge it, phone started constantly rebooting. It would stay on as long as I would keep using it. But if I put it down put a couple minutes, it would restart.
Also, could not make any calls/receive messages. Sim card works fine in my SIII, so it is not a bad sim card. The phone just won't recognize any sims.
First time this happened was after a few months. I tried wiping the phone completely, but when it came time to set it back up, it still did not recognize the sim, so I couldn't get past the activation screen.