There are three examples I can think of:
Shooting a portrait in bright sunlight. If you have a wide aperture lens, with something like F/1.4, and there is a lot of light, it could be that the fastest shutter speed of your camera (1/4000 or 1/8000 for example) is not sufficient. This means that the picture will be over-exposed, because you want to achieve a narrow depth of field in this brightly lit scene. Using a ND-filter can solve this problem. See an example here, how the DOF can be narrowed by a ND-filter.
Another example, but this is not very common, is to be able to do panning shots of for instance car racing. For this you also have constraints on your maximum shutter speed. By using a ND filter, you can create panning shots, with lots of movement AND a narrow depth of field. Like in this example at F/1.8: (clickable to its original source)
3, Creating a narrow depth of field in your shot while using a flash. If you are constrained by the sync speed of the flash, you have to increase your aperture number. If you still want to achieve a blurred background you can use a ND filter.
So in all cases the artistic effect which will be achieved is the shallow DOF itself. But in these situations, when you have constraints on the shutter speed, you will achieve this thanks to the ND filter.