Increasing the smoothness of a mesh object does not increase the mesh cell count, as shown in the following diagram. The number of cells is the same, but the cell edges and surfaces are more curved with the increased smoothness:

If a sufficient number of cells is present, you can have the same smooth body, but this time represented by cells with straight edges and flat surfaces. This operation is called refining the mesh object. The Refine Mesh tool button can be found in the Edit panel on the Home tab in the 3D Basics workspace, or in the Mesh panel on the Mesh tab in the 3D Modeling workspace, as shown in the following screenshot:

The refine mesh command works only with smoothed objects. If the object already has straight edges and flat cell faces, then there is no point adding more cells. To refine a mesh object, simply click on the Refine Mesh button and then click on the desired mesh object to be refined. The following diagram shows an example of an object that is created with no smoothness, then with the smoothness increased once, and then with the mesh refined:

The previous diagram shows an example of an object being smoothed, and then, in the next refining step, more divisions were added to accommodate the smoothness added in the previous step.