It appears that the September 2013 update of Office breaks file associations on some machines. Double-clicking a file will result in an error message. Opening the relevant application and then clicking File – Open still works as usual.
Users with Office 2010 Starter seem to be the hardest hit: when they try to repair Office, they are now offered a 30 day trial of Office 365, which many users are not happy about. Understandably! 😉
MS Word MVP Suzanne Barnhill has written up a diagnosis and some workaround suggestions for 2010 and 2010 Starter on the Microsoft Community site here.