That is not true. Well, not entirely. Gift box have pretty much "unlimited" capacity, but
1) You will only see your 100 top items. Ones pushed beyond this window are still there, but will not be accessible until you remove enough top items to bring them in that last 100
2) All items in present box have expiration date of 1 month since acquisition. So if you let them lie in your box for too long, they will rot and disappear)