Microsoft has now responded to the issues Xbox Live subscribers are experiencing. Apparently the specific problems users are having have to do with the Xbox Marketplace, which is where people can purchase and download playable game demos, Xbox Live Arcade games, TV Shows, and more.
Here is Microsoft's statement regarding the new technical issues.
"We are aware of the problem and working to resolve the technical issues. We apologize for any inconvenience this may cause and thank you for your patience." -Microsoft