This paper is concerned with the finite-time state observer design for robotic systems with lossy state measurement and nonlinear dynamics including uncertainties and disturbances. The extreme learning machine (ELM) algorithm is employed to approximate the nonlinear dynamics, and simultaneously, adaptive technology is applied to regurate the output weights of the ELM network and to remove the adverse effects of residual errors and disturbances. Then, a finite-time state observer on the basis of the adaptive signals is proposed to approximate the immeasurable states within a finite time accurately. Ultimately, the estimation accuracy of the designed finite-time ELM network-based observer is demonstrated by simulation results on a robotic manipulator platform.