When you think you have come up with a great idea or response / reply to something, but you have only half-thunk it, and therefor you have a broken thought. Here's a good example.
Announcer: Question number 13, What are several things people are doing these days besides getting involved with tech?
James (in his head): " Oh this is gonna be easy! I can think of several already! " *BUZZ!*
James: " Well, people do lots of things each day like... " (in his head) Oh shit... I can't think of anything...
Announcer: Tiiiimes Up!
(James, and D-Jay, after the contest)
D-Jay: What happened back there man? Cat got your tounge?
James: I had a broken thought.