Resolvable Balanced Incomplete Block Designs (RBIBDs) are important combinatorial designs that have useful applications in various fields of human endeavour. In this paper, Nim addition tables, from the game of Nim, of order 2ⁿ, 2≤n≤5 were used as the basis for the construction of some (k², b, r, k, 1) RBIBDs. Nim addition tables of order 2ⁿ, 2≤n≤5 were constructed. These tables were special Latin squares that obeyed the Finite Groups theory and closed n-nim-regularity conditions for closed n-nim-regular games. The Bose’s generalized method of constructing Mutually Orthogonal Latin Squares (MOLS) was used to obtain 2ⁿ-1 MOLS for each n. The MOLS were super-imposed on one another and successive diagonalization algorithm was used to obtain the RBIBDs from the super-imposed MOLS. The RBIBDs constructed were (16, 20, 5, 4, 1), (64, 72, 9, 8, 1), (256, 272, 17, 16, 1) and (1024, 1056, 33, 32, 1) RBIBDs. These RBIBDs are all in existence and a link was thus established between the game of Nim and RBIBDs.