That's actually happened more than once to a friend of mine. He has the same name, Michael James Smith, as a local repeat offender. He was once denied the lease on an apartment because of it, and by the time it was sorted out, the place had been rented to someone else. If he gets pulled over for a traffic violation, it takes the cops forever to make sure he's not the Michael James Smith they always seem to have an arrest warrant for because, as bad luck would have it, they also look a lot alike. It's a big hassle.
If I had a common last name, I'd never pick a common first name (especially since I wouldn't want to misspell it to make it different!). One thing I like about my super-rare last name is that I feel the whole popularity spectrum is open to me.