I have set the mouse to be a custom cursor under Edit > Project Settings > Player. (I set the cursor with script originally and had the same problem.) In the editor it shows up as the system cursor. When I run my game it shows the custom cursor for only as long as I don't move the mouse or click a button.
But, interestingly, if I hold down the middle mouse button in the editor it will show up as a custom cursor! Any ideas? I have the texture set to cursor, clamp, point, and read/write is enabled.