Let omega be any linear symplectic form on the 4-torus T-4. We show that in all cases. (T-4,omega) can be fully filled by one symplectic ball. If (T-4,omega) is not symplectomorphic to a product T-2(mu) x T-2(mu) of equal sized factors, then it can also be fully filled by any finite collection of balls provided only that their total volume is less than that of (T-4,omega).