I'm learning the basics of Cycles and I created a simple cup with a material with a Glossy BSDF shader.
The scene is enclosed by 6 planes and I tried illuminating with a point lamp and then with a light emitting plane, but the shadows remain. Below you can see the render and the mesh.
The rendering settings are the default ones except for the number of samples.
After examining the mesh closer I realized the problem might be there, the subdivision surface tends to keep the slope of the straight edges of the original mesh. The first picture below shows the render and the mesh (after applying the subsurf) from the same point of view. The second one shows the original mesh, which was made with the spin tool. I tried beveling but the problem persisted.
The blend is here: file.blend