I also have had this issue on multiple devices since upgrading to iOS 6. It is still a problem in 6.0.1. The devices I have this problem with include an iPhone 4, an iPhone 4S, two iPad 2s, and an iPad mini.
I called Apple support this morning. After going through the expected steps of resetting my network settings and finding that the problem remained, the agent put me on hold for a few minutes. When she returned, I was told that this was a known issue. It is systemic and not isolated to particular Apple ID's. She stated that a resolution was being worked on but could not share when it would be available. In the meantime, she recommended that I sync with iTunes to view and install former purchases.
I politely explained that this is simply not an option for me. Once I sync a device with iCloud, and then try to sync with iTunes again, I lose work and end up restoring from backup. This has happened on multiple occasions and on multiple devices. She apologized and wished me good health and a long life.
I have been a faithful Apple customer for years. In addition to the devices listed above, our household includes a 15" MacBook Pro, a MacBook Air, and a Mac Mini.
I am quickly losing faith in Apple's "It Just Works" tag line. iOS 6 has been an embarrassment from maps to App Store crashes. I hope that someone from apple sees this and escalates these types of issues. We need Apple products to regain their reliability and usability if we are to remain faithful customers.