It's possible you might be thinking of:
* The PC Box System: In Pokémon Diamond, you can store Pokémon in Boxes within the PC, similar to a storage system. However, there's no specific "Trainer Case" feature.
* The Item Storage: You can store items in your Bag, but there's no specific "Trainer Case" for holding items.
If you can provide more context about what you're looking for, I might be able to help you find the correct information!