Here is a strange issue (for the users, but not really) where a user having Edit permissions on a particular document receives an error 'Access Denied' when editing the Name site column/field of the document (the Name column value is automatically captured from the document name).
Strange because the user actually has required document edit permissions but still cannot edit. In a SharePoint list there is a setting to allow users to delete only their owned items, but there is no such settings for document library.
Interestingly, SharePoint requires Add, Edit and Delete permissions for a user to edit the document Name in the document library. Editing the Name value will fail if any of the three permissions are missing. So make sure the user has delete permissions in order to modify the document name.
This could happen in following common cases (assuming Edit permissions are granted) -
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