I've noticed that while playing rhythm games--games where I must press buttons precisely in step with a musical accompaniment--I can either play or I can talk. If start talking, I'll almost immediately lose the beat, or I'll talk between the gaps in the beat in a very odd but unintentional manner; it's like I simply can't focus on both at the same time.
I (and many others, judging from Twitch.tv) can still input precise button processes at exact times while talking, so it's not my sense of timing that's off. And it's not that my talking covers the music; I can turn the music up while talking and still falter, and I can mute the music after I get going and keep up the beat. It seems to be specifically my rhythmic timing that's off when I speak.
What would cause speech to interrupt one's sense of rhythm like this when it doesn't interrupt other types of precise timing? I was unfortunately quite unable to find anything about this while searching, just a bunch of articles on impaired speech.