  Motivated by the notions of $2$-good rings and fine rings, we define a
$2$-nilgood ring as a ring whose every non central-unit is the sum of two
nilpotents. This is a study of $2$-nilgood rings. The following results are
obtained: (a) a ring is $2$-nilgood iff it is either a $2$-nilgood simple ring
or a commutative local ring with nil Jacobson radical; (b) a $2$-nilgood simple
ring is a field if it is right Goldie; (c) a $2$-nilgood ring containing a
uniform right ideal is a commutative local ring with nil Jacobson radical; (d)
a ring whose every non central-unit is a sum of a square-zero element and a
nilpotent is a commutative local ring with nil Jacobson radical. We conclude
with the conjecture that a $2$-nilgood simple ring is a field.
