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In this paper, the problem of finite-time attitude-tracking control for a rigid spacecraft is addressed. Uncertainties including unknown inertial parameters, external disturbances, actuator failures and saturation constraints are considered. Firstly, a smooth function which is different from the common saturation treatment is presented to deal with the actuator constraints. Secondly, a fast non-singular terminal sliding mode (FNTSM) manifold composed of tracking errors is constructed. To estimate the unknown function in the sliding surface derivative, Chebyshev neural network (CNN) is introduced and thus the strict assumptions on uncertainties in many related works are abolished. By designing the CNN adaptive laws, a new fault-tolerant control scheme is proposed such that the attitude tracking can be achieved within a limited time interval. Compared with the existing CNN-based achievements with finite-time convergence, the approximation errors are proved to be finite-time stable instead of uniformly ultimately bounded (UUB). Finally, simulation experiments are conducted to demonstrate the satisfactory tracking performance of the attitude controller. [ABSTRACT FROM AUTHOR]