This is like asking: "Can I cook both the entree and dessert?"

At a large restaurant (a game development studio) there will certainly be different people for each role. At a small, mom&pop restaurant (amateur/indie game developer) there might just be one person fulfilling both roles.

Some people are good at one or the other. Some people are excellent at both. Typically, if you can afford it, you'd prefer putting a specialist in each role, so that he/she can concentrate on one type of task and so that his/her on-the-job learning will be very focused. If you can't afford it, then you don't have an option. It's either do both or do neither.

I think being both programmer and designer is actually the ideal setup. It gives the designer a more intimate intimate relationship with the medium (and as a programmer, it's much more fun making my own stuff!).