In this talk we focus on the uniform convergence rates from nonlocal models to the corresponding local models, and presents a necessary condition to guarantee the first-order and second-order convergence rate with respect to a nonlocal horizon parameter $\delta$ without extra assumptions on the regularity of nonlocal solutions. To do so, we first revisit the maximum principle for nonlocal models, and present the uniqueness of the nonlocal solutions. After that, we give the methodology to address the truncated errors on the volume constrains or Neumann BCs, and then combine the resulting errors from boundary layers with the maximum principle to obtain the uniform convergence order. Our analysis shows that the constant value continuation of the boundary conditions of local problems only leads to first-order convergence rate. And if we expect to have second-order convergence rate, the information of first-order derivatives for local problems on the boundaries is required. One and two dimensional numerical examples are given to verify the effectiveness of our theoretical analysis.