In this paper, we study sum formulas for Schur multiple zeta values and give a generalization of the sum formulas for multiple zeta(-star) values. We show that for ribbons of certain types, the sum over all admissible Young tableaux of this shape evaluates to a rational multiple of the Riemann zeta value. For arbitrary ribbons with $n$ corners, we show that these can be always expressed in terms of multiple zeta values of depth $\leq n$. In particular, when $n=2$, we give explicit, what we call, bounded type sum formulas for these ribbons. Finally, we show how to evaluate the sum over all admissible Young tableaux with exactly one corner and also prove bounded type sum formulas for them. This will also lead to relations among sums of Schur multiple zeta values over all admissible Young tableaux of different shapes.