If you have an old tree stump in your garden, then you may have decided to let it rot down naturally. If the stump isn't in your way, then it may not bother you too much right now.
However, tree stumps don't necessarily rot away quickly. Even small stumps can last for months, and large stumps may take years to degrade on their own.
Plus, if the tree stumps roots are firmly in the ground, they may still be supplying the stump with some nutrients.