The finite-time containment control problem of high-order nonlinear multi-agent systems with double input constraints is investigated in this paper. Different from the existing literature on the multi-agent systems with input constraints, this paper considers the finite-time compensation of both unknown dead zones and random stuck faults, and does not set a restrictions on continuous uncertain nonlinearity. To address nonlinearity and avoid singularity in recursive design, both fuzzy logic systems and adding a power integrator technique are introduced. Then, bound estimation method is integrated into adding a power integrator technique, so that two novel finite-time compensation laws are designed to achieve the rapid compensation of unknown dead zones and random faults. Finally, a novel finite-time fuzzy control scheme is developed to achieve finite-time containment with double input constraints, and which is rigorously proved and verified by comparison experiment and application example.