The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$u_t=v^p\left(\Delta u+a{\int }_{\Omega }u\mathrm{d}x\right),v_t=u^q\left(\Delta v+b{\int }_{\Omega }v\mathrm{d}x\right)$$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution $(u,v)$ to this problem. Moreover, a necessary and sufficient condition for the non-global existence...