I'm having the exact same problem, and it's quite annoying because I now have to press the home button on my ipad 3 about 20 times before I can get it to respond, and even then it freezes up. It seemed to have started after upgrading to iOS6, which is really kind of disappointing because you guys (at Apple) charge a lot of money for these devices and create quite a build up prior to releasing new features promising all these new/improved features, but there's always a flaw that overshadows the goo things isn't it? Shouldn't you guys have tested all these things prior to giving us some faulty software and making us go out of our way to try and get these things sorted out for us at apple stores or whatever? I bought my ipad three not two months ago, paid $700 for, and already it's starting to give me issues. I had a similar problem with my iPhone 4, when maybe after you released iOS5 the home button stopped working. I had to go to the store and they couldn't fix it so the solution was to mail me a new device and return the faulty one to Verizon. And when I got the new device, I had to go through the headache of resyncing all my contacts, music, apps, and some features weren't working either. Like for when I tried to sync playlists, I couldn't transfer all the music into the new device because of something to do with protected files purchased from iTunes. I had to do a bunch of research before I could figure out how to actually get my music to sync on the phone.
Now I feel like I'll be in a similar position where, if I go to the apple store, they won't be able to fix this problem and they'll make me mail my device to apple for an exchange for a new one and I'll have to go through the same process again.
If you guys want customer satisfaction, how about you deliver us with products that actually work as they're intended to instead of forcing us to have to do all this extra stuff? It's not like you're going to be willing to return the money I've spent on your faulty product if I've gotten fed up with it are you?