Interesting that most replies are centred around film and TV. What about the stage? There are a lot of stage productions that use music (as opposed to musicals which are a different kettle of fish). My first introduction to this was "The Mysteries" back in the early 80s. The obvious name from that production was John Tams who is a fine musician and a good actor. I think you wil find that the crossover of actors and musicians has been much more widespread on the stage, but of course the ephemeral nature and scale of stage productions means that fewer people see them and there is no lasting record.
In those days there was no formal training available (I think) but that is now changing. My son Jared has recently graduated from Rose Bruford college with an actor/musician degree. He is a fine actor and also a talented multi-instrumentalist. There are many more like him and the theatre is changing with more and more productions using music as an integral part of the production i.e. the run of Brecht's "The Resistable Rise Of Arturo Ui" based at the Watford Palace that has just finished. Hopefully the right people will get the jobs in future but I doubt it while Hollywood feels that a big name is better than acting talent!