The m-bonacci word is a generalization of the Fibonacci word to the m-letter alphabet A = {0,..., m - 1}. It is the unique fixed point of the Pisot-type substitution phi(m) : 0 -> 01, 1 -> 02,..., (m - 2) -> 0(m - 1), and (m - 1) -> 0. A result of Adamczewski implies the existence of constants c((m)) such that the m-bonacci word is c((m))-balanced, i.e., numbers of letter a occurring in two factors of the same length differ at most by c((m)) for any letter a is an element of A. The constants c((m)) have been already determined for m = 2 and m = 3. In this paper we study the bounds c((m)) for a general m >= 2. We show that the m-bonacci word is ([kappa m] + 12)-balanced, where kappa approximate to 0.58. For m <= 12, we improve the constant c((m)) by a computer numerical calculation to the value [m + 1/2].