Can an ionic compound ever consist of a cation-cation or anion-anion bond?

No, it is impossible. By definition, an ionic compound is a combination of two elements, one of which is a metal while the other is a nonmetal. Together, they form an ionic bond, which is a complete giving and accepting of electrons to satisfy their full octet individually. A cation-cation or anion-anion bond is a different type of bond called covalent bond. This involves sharing of electrons.