copy/pasting between projects is not the way to go.
If you wish to nest a project under a different project, you may open the projects module in grid mode, and there hover your mouse cursor on the case with the project name. This will display the small edit icon on the right, which you can click. There, you will be able to change the parent of the project, which you can then set to the project of your choise.
Your original project will now be a sub project of the one you select.