  In this paper, we prove that a compact quasi-Einstein manifold
$(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative
sectional curvature and zero radial Weyl tensor is either isometric, up to
scaling, to the standard hemisphere $\Bbb{S}^n_+,$ or $g=dt^{2}+\psi
^{2}(t)g_{L}$ and $u=u(t),$ where $g_{L}$ is Einstein with nonnegative Ricci
curvature. A similar classification result is obtained by assuming a
fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is
presented in order to justify our assumptions. In addition, the case of
dimension $n=3$ is also discussed.
