We apply collisionless particle-in-cell simulations of relativistic pair plasmas to explore whether driven turbulence is a viable high-energy astrophysical particle accelerator. We characterize nonthermal particle distributions for varying system sizes up to\ L/2πρ_{e0}\ =\ 163, where\ L/2π\ is the driving scale and\ ρ_{e0}\ is the initial characteristic Larmor radius. We show that turbulent particle acceleration produces power-law energy distributions that, when compared at a fixed number of large-scale dynamical times, slowly steepen with increasing system size. We demonstrate, however, that convergence is obtained by comparing the distributions at different times that increase with system size (approximately logarithmically). We suggest that the system-size dependence arises from the time required for particles to reach the highest accessible energies via Fermi acceleration. The converged power-law index of the energy distribution,\ α\ ≈\ 3.0 for magnetization\ σ\ =\ 3/8, makes turbulence a possible explanation for nonthermal spectra observed in systems such as the Crab Nebula.