Take $a_{iiii}=0$ and let the other $a$'s be arbitrary large. Then the inequality is satisfied.
Indeed, let $w_1, \dots, w_{17}$ be arbitrary nonnegative integers. Without loss of generality assume that $$\min \{ w_1, \dots, w_{17} \} = w_1.$$ Then $$\min_{1\leq i\leq j\leq k\leq l\leq 17} (a_{ijkl} + w_i + w_j + w_k + w_l) \leq a_{1111} + 4w_1 = 4w_1 \leq \frac{4}{17} \sum_{t=1}^{17} w_t$$ as required.
Hence, $a_{ijkl}$ are not bounded in general.