To wedge oneself in between people or to cut into a conversation or quarrel; to push in or intervene.
二人が話しているところに彼が割って入った。
He cut in while the two of them were talking.
バス停で割って入るのはマナー違反だ。
Pushing in line at the bus stop is bad manners.
けんかの仲裁に割って入ると、事態を収めやすくなることもある。
If you step in to mediate a fight, you can sometimes help calm things down.
会議で他人の発言に割って入る前に、発言の順番を確認しよう。
Before interrupting someone in a meeting, check the speaking order.