Some object types enforce rules relating to the uniqueness of their name. In the case that an object type enforces unique names, an error message will be shown asking the user to select a unique name if the name entered is the same as an existing object name. For example if you have named an asset "Test Asset", you cannot give that name to another asset.
Some object types enforce uniqueness on a per account basis. A good example of this is the user. A user's name must be unique within an account.