Can potential energy be considered a special case of internal energy? Why or why not?

No, potential energy cannot be considered a special case of internal energy.
Potential energy is the energy acquired by a body due to change in its position or its configuration. Internal energy is the sum of the kinetic energy associated with the random motions of the atoms or molecules. It is associated with the forces between the atoms or molecules. So, the internal energy of a system includes the microscopic kinetic and potential energies of the atoms or molecules of the system.
But there are other forms of potential energies such as gravitational potential energy and elastic potential energy which are not internal energies.
Therefore, potential energy cannot be considered a special case of internal energy.