SOLVED!!
Well, I had the same issue with my iPad2 from the moment I got it home; had severve problems with iTunes / AppStore, and the connection to YouTube and GameCenter failed frequently. I phoned support, but all they came up with was a reset of the iPad, which then was OK for 5 minutes before things started to fail again.
But no matter how many connections failed, I was always able to browse the internet through Safari, and I could ping the iPad from my pc, so there was a internet link. And all my other WiFi-gadgets was working flawlessly; iPhone, internet radio, VoIP-telephone...
When I then took my iPad to work, and hooked it up to our guestnet, I had no problems with AppStore or iTunes, so I then began to think that my WiFi-router was the culprit. So I went to the store and bought a Cisco/Linksys router, on the condition that if my iPad still wouldn't connect, I'd get to return the router.
Now it does not appear that the store will get the Cisco router back anytime soon... The iPad has been working flawlessly ever since I swapped my Belkin router (2 yrs old) for the Cisco router.