Your plan decides it. In My designs & templates, the line under Save this design always shows how many designs you have saved and how many your plan allows. If your plan has no limit, the line shows only how many you have saved.
When you are at your limit, nothing is deleted. The panel says you have saved all the designs your plan allows, and asks you to choose one to replace or to upgrade. To replace one, press Use this slot on a design you no longer need, then Replace and save. To keep more, upgrade your plan under Subscription Plan in the side menu.
The design you are working on cannot be replaced or deleted. If that is the one you want to let go, open another design first.
Remember: when you apply a new theme in the Theme Store and press Save and apply, your current work is saved as a design, and it counts toward your limit like any other.