For mixed-integer linear and nonlinear optimization problems we study the objective value of feasible points which are constructed by the feasible rounding approaches from Neumann et al. (Comput. Optim. Appl. 72, 309–337, 2019; J. Optim. Theory Appl. 184, 433–465, 2020). We provide a-priori bounds on the deviation of such objective values from the optimal value and apply them to explain and quantify the positive effect of finer grids of integer feasible points on the performance of the feasible rounding approaches. Computational results for large scale knapsack problems illustrate our theoretical findings.