We give a complete description of the primary degenerations and non-degenerations between the 3-dimensional nilpotent algebras, the 4-dimensional nilpotent commutative algebras and the 5-dimensional nilpotent anticommutative algebras over C. It follows that all the varieties under consideration are irreducible and determined by the rigid algebra N₂, the family of algebras C₁₉(α) and the rigid algebra A₁₁, respectively. In particular, as an auxiliary new result, we obtain an algebraic classification of the 4-dimensional complex nilpotent commutative algebras.