How can I limit the maximum number of items allowed for the ENTIRE CART regardless if the items were from the same product or from different products?
Magento default just allows you to set up maximum quantity allowed in the shopping cart of a product only. Therefore, I think that you had better look for an extension to help you overcome this issue quickly.
For example, you can try this extension
It can help you to set up both minimum and maximum order quantity of many products that belong to a category.
In your case, because you want to limit the maximum number of items allowed for the ENTIRE CART regardless if the items were from the same product or from different products, so using this extension, you can add many products into a category and then set up maximum order quantity for that category.
Finally, you can limit maximum numbers of products allowed in the shopping cart
I think this is useful for you.
Admin -> store-> Configuration -> catalog -> Inventory -> Product Stock Option
Change your "Maximum Qty Allowed in Shopping Cart" then, save and remove var/* then check it.
You can use below extension for this. It's meet your requirements.