For fixed weights w_1,...,w_n, and for d>0, we let B denote a collection of d*n balls, with d balls of weight w_i for each i=1,...,n. We consider the problem of assigning the balls to n bins with capacities C_1,...,C_n, in such a way that each bin is assigned d balls, without exceeding its capacity. When d>>0, we give sufficient criteria for the feasibility of this problem, which coincide up to explicit constants with the natural set of necessary conditions. Furthermore, we show that our constants are optimal when the weights w_i are distinct. The feasibility criteria that we present here are used elsewhere (in commutative algebra) to study the asymptotic behavior of the Castelnuovo-Mumford regularity of symmetric monomial ideals.