Note from Mojang:
Non-1.7 clients are currently getting Failed: Bad Login error when trying to log into multiplayer servers. 1.7 clients are not affected. There is no ETA for a fix. Please wait patiently as Mojang fixes the issue.
As a result of this, I will update the server temporarily to Minecraft 1.7. This version WILL NOT HAVE ANY Minecraft 1.7 features. It is simply a Minecraft 1.6 server which accepts Minecraft 1.7 clients.