I guess that in the real world, you will have 2 halves and a hinge, right? In your model, you are only making a cut into one side of the ring - when you do that in the real world, that ring will still be one solid with a fracture on one side.
Consider this picture
Your top and bottom of our main object is basically a square piece with two circular cuts - like the object on the right in the picture. When you try to make a cut with the blue line, the thing will still hang together.
If you make your geometry by revolving a line around a center point, you will end up with geometry like on the left side in the picture. If you now use the same blue line, the surface will split because there already is an edge in there.