We consider Integer Programs (IPs) where each variable corresponds to an integral point within a polytope $\mathcal{P}\subseteq \mathbb{R}^{d}$, i.e., IPs of the form $\min{c^{\top}x\mid \sum_{p\in\mathcal P\cap \mathbb Z^d} x_p p = b, x\in\mathbb Z^{|\mathcal P\cap \mathbb Z^d|}{\ge 0}}$. The distance between an optimal integral solution and an optimal fractional solution (called the \emph{proxmity}) is an important measure and a classical result by Cook et al.~(Math. Program., 1986) shows that it is at most $\Delta^{\Theta(d)}$, where $\Delta=\lVert \mathcal{P}\cap \mathbb{Z}^{d} \rVert{\infty}$ is the largest coefficient in the constraint matrix. Another important measure studies the change in an optimal solution if the right-hand side $b$ is replaced by another right-hand side $b’$. The distance between an optimal solution $x$ w.r.t.~$b$ and an optimal solution $x’$ w.r.t.~$b’$ (called the \emph{sensitivity}) is similarly bounded, i.e., by $\lVert b-b’ \rVert_{1}\cdot \Delta^{\Theta(d)}$ as shown by Cook et al.~(Math. Program., 1986). Even after more than thirty years, these bounds are essentially the best known bounds for these measures. While some lower bounds are known for these measures, they either only work for very small values of $\Delta$ or require negative entries in the constraint matrix. Hence, these lower bounds often do not correspond to instances from algorithmic problems. In this work, we present for each $\Delta > 0$ and each $d > 0$ IPs of the above type with non-negative constraint matrices, such that their proximitiy and sensitivity is at least $\Delta^{\Theta(d)}$. Furthermore, these instances are closely related to instances of the Bin Packing problem as they form a subset of columns of the \emph{configuration IP}. We thereby show that the results of Cook et al.~are indeed tight, even for instances arising naturally from problems in combinatorial optimization.