iCloud system status. | Source: Apple
According to a reliable source, Apple has sent out an internal P1 (high priority) alert notifying employees of server issues that are preventing customers from activating their iPhones, whether the device is brand new or recently restored. Compounding the problem are "other internal issues," though the exact nature of these complications is unknown at this time.
Since iOS 7 launched on Wednesday, Apple's servers have been bombarded with a deluge of users looking to update to the latest operating system, with many finding the activation service unavailable or otherwise unresponsive.
The alert failed to offer the exact number of users affected and did not give an estimate as to how long the activation server would be down.
Apple's System Status webpage shows all services are up and running as of this writing, though AppleInsider has received numerous reports that appear to corroborate the backend problems.