I'm experiencing this again for the 2nd time with my iPad 2. The first time I just figured it was broken and put it away for several weeks (maybe longer) and one day I thought I'll try one more time and voila it magically worked. Fixed itself. I was able to sync it to my laptop again as well.
Now about 1 year later it's back again. Same exact problem. If I power off and plug it in it turns itself back on but won't charge. So I have to plug it in first and then power down and let it charge (painfully slowly). We'll see if time will heal this again :\