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A ring is defined to be 2-nil-good if every element in is the sum of two units and a nilpotent. Fundamental properties of such rings are obtained. We prove that every strongly -regular ring is 2-nil-good if and only if the identity is the sum of two units. One of the main result of this paper is that every square matrix ring over J-fine rings is 2-nil-good. We establish 2-nil-good property for Morita contexts. This implies, in particular, that every matrix ring over 2-nil-good rings is 2-nil-good. Furthermore, we prove that the ring of all lower triangular diagonal-finite matrices over a 2-good ring is 2-nil-good.