Not interrupting isn't always the polite choice, sometimes it just means a meeting runs long, a decision gets made without a key piece of information, or the same point gets repeated for the third time because nobody redirected it.
The discomfort around interrupting usually comes from treating all interruptions as equally rude. They're not. An interruption that serves the room, redirecting a conversation that's stalled, surfacing time-sensitive information, keeping a meeting on track, reads completely differently from one that just serves the interrupter's need to be heard. The skill isn't avoiding interruption entirely. It's interrupting for the right reasons, in a way that's clearly in service of the group.
"Sorry to jump in, I want to make sure we don't lose the thread on the budget question before we move on, since we're at time." That's a redirect in service of the meeting, not a grab for airtime, and most people register it that way even in the moment.
Tone carries most of the weight here. A brief acknowledgment, "sorry to jump in," "can I add something quickly", signals that you're aware you're interrupting and doing it deliberately, not carelessly talking over someone. Skipping that signal, even with good content to add, tends to land as inconsiderate regardless of how useful the point turns out to be.
People rarely resent being interrupted for a good reason, said with respect. They resent being talked over without either.
Timing matters too. Interrupting mid-sentence, before someone's finished a thought, reads very differently from waiting for a natural pause, even a half-second one, and then jumping in. The content of the interruption can be identical; the difference in reception is almost entirely about whether the other person felt cut off or given a clean handoff.
One category worth naming specifically: interrupting to redirect a meeting that's run long or gone off-topic. This is one of the most valuable interruptions a first-time manager can learn to make, and one of the most avoided, because it can feel presumptuous to redirect a conversation you're not formally running. A simple, neutral version, "we've got ten minutes left and haven't covered the budget question yet, should we come back to this later?", does the job without anyone feeling overridden.