> watched the tv and listened to the radio

I am fine with your other examples but do people really do this?

I always had the tv OR radio on while doing other things, but the idea of having both on at the same time seems incredibly strange to me.

At least for sports, muting the TV and listening to the radio broadcast of the same game can greatly improve the experience. Muting the weather channel and a few others is already a net gain even without the radio.

Not OP but I think that the comment about both TV and radio on at the same time was more generic than that.

I'd argue that it's a known fact, at least for anybody that follows any mainstream sport, that sports radio commentary is a lot more exciting that its tv counterpart.

Right, I was surprised by the idea one could be listening to two different things at once, not mixing different audio/video streams.