In a bounded domain \begin{document}$\Omega\subset\mathbb{R}^n$\end{document} , where \begin{document}$n\ge 3$\end{document} , we consider the quasilinear parabolic-parabolic Keller-Segel system \begin{document}$\begin{equation*}\begin{cases}u_t = \nabla\cdot({D(u)\nabla u+u\nabla v}) \;\;\; &\text{in}\ \Omega\times(0,\infty)\\v_t = \Delta v-v+u &\text{in}\ \Omega\times(0,\infty)\end{cases}\end{equation*}$ \end{document} with homogeneous Neumann boundary conditions. We will find that the condition \begin{document}$D(u)\geq Cu^{m-1}$\end{document} suffices to prove the uniqueness and global existence of solutions along with their boundedness if \begin{document}$D(0)>0$\end{document} and \begin{document}$m>1+\frac{(n-2)(n-1)}{n^2}$\end{document} which is a very different result from what we know about the same system with nonnegative sensitivity. In the case of degenerate diffusion ( \begin{document}$D(0) = 0$\end{document} ) and for the same parameters, locally bounded global weak solutions will be established.