According to Iwata-San the N5 (Nintendo's codename for the next console) will be able to play GameCube games.
In an interview with Famitsu, the head of Nintendo Satoru Iwata has stated the N5 will have backwards compatibility.
"It will be able to use the games already created in the previous generation. And it is something that will work. Side performances and graphics, one reached today a kind of limit with a quality which is already close to realism and it will be difficult to still much better do. The role of Nintendo is thus to study other ways in order to improve the experience of the player."
News source: GameCube Europe