Apple’s official statement:
We experienced a brief issue with the software update process, affecting a small number of users during the first hour of availability. The problem was quickly resolved and we apologize to those customers. Anyone who was affected should connect to iTunes to complete the update or contact AppleCare for help. – Apple Inc.
MacDailyNews Take: Issues happen. Thankfully, Apple quickly sprang into action and resolved the issue before it impacted significant numbers of users. All of our iOS 10 updates were flawless (we do use iTunes to update whenever possible).