This article focuses on an adaptive neural network (NN) finite-time prescribed performance control problem for nonstrict-feedback nonlinear systems subject to full-state constraints. Specifically, a finite-time performance function is employed, which can guarantee that the tracking error converges to a prescribed region within a finite-time. Neural networks (NNs) are used to approximate the unknown nonlinear function. The unmeasurable states are estimated via constructing a state observer. By using the dynamic surface control (DSC) technique, the complexity problem has been avoided in traditional backstepping control. In order to satisfy the state constraint condition, the barrier Lyapunov function (BLF) is incorporated in the process of backstepping. The developed adaptive finite-time NN backstepping control strategy can make that the closed-loop system is semiglobally practical finite-time stability (SGPFS). Meanwhile, all states can be guaranteed to remain in the constrained space. Simulation results demonstrate the validity of the control method.