Table design for departments with duplicate department numbers
The PK of your department table is not dept_id.  The column dept_id is an attribute of a manager table that you have not defined, since it is functionally dependent on the manager_id.

The PK of your department table is dept_name.

Primary keys don't have to be integers.

Note: If your HR department creates departments with the same name, that is going to land you back in hot water.  You might need to create a composite primary key of dept_id and dept_name.  This could still lead to trouble if your HR department can't be trained to keep department names unique within managers.  Excel is not a good data source for being able to enforce constraints.