Hey,
I had the exact same issue. Delete the account from system preferences. Shut down the computer (Control,Option,Command,Eject), then turn it back on (don't just restart). Add the account back into System Preferences and it should work. Well, at least it worked for me.
If for whatever reason that doesn't work, try the usual suspects: Disk Utility (Click Repair Disk Permissions), Twitter.com (Reset the Password), Log out and Log into your user account, etc.
If it still isn't working, perhaps there's an app conflicting with it. Check your menubar for 3rd-party apps that are running all the time, try turning them off one at a time and try sending a tweet then.
These kinds of things fix at least 3/4 of the problems out there.
Good luck!