The angular momentum operators $L_x = yp_z-zp_y$, $L_x = zp_x-xp_z$, $L_x = xp_y-yp_x$ do not commute. In fact, $[L_x,L_y] = i\hbar L_z$. We also know that $L_z |nlm\rangle = \hbar m$. So what if we have an eigenstate $\psi = |100\rangle$, this will make $L_z \psi = 0$. For this state, can we measure $L_x$ and $L_y$ simultaneously, in theory?