Can't add custom fields to custom content types

When I create a custom content type, the only fields available are Name, Type, and Description (not counting the Submission, Workflow, and Comment settings menus). There is no link to add new fields.
If I save that content type and access it later, the only option is to "edit" or "delete" -- edit takes me back to the same screen where all I can edit is Name, Type, and Description.
As far as I know, I do not have to add a CCK to Drupal 6.8 in order to add basic, custom fields (text, select, drop-down, etc.). Any help?