Object Follows the Cursor, But Snaps to Grid

I’m trying to make a game that looks low-resolution, and thus I’m upscaling everything by 2x. However, the game is to have a custom cursor, and it ruins the illusion if the cursor follows your mouse every pixel. I want the cursor to follow your mouse, but to snap to every two pixels so as to not ruin the illusion of low resolution. This is possible, right?