I would hazard a guess that the "the break of 10:30" is the first coffee break of an eight-hour shift at a factory.

I'm guessing this has to do with radio or television. You can correct me if I am wrong. On commercial radio and television there is usually a break (pause) every thirty minutes to either change shows and/or broadcast a commercial. In view of that, I would guess that the speaker is talking about the break that was supposed to occur at 10:30 in the programming. Apparently, something did not go smoothly.

break = pause; rest period