Precise measurements of radial (RV) and rotational (vsini) velocities of stars are essential for studying stellar kinematics (space velocities and dispersions), binary orbits (mass measurements and formation), and rotational dynamics (angular momentum evolution). The high-resolution spectroscopic observations necessary to make these measurements are challenging for the intrinsically faint and low-temperature ultracool dwarfs, stellar and sub-stellar objects with masses below 0.1 MSun. Previous local UCD kinematic studies indicated conflicting L dwarf kinematic ages, with little constraints on the T dwarf kinematic due to the small sample size. We present a radial and rotational velocity survey of 37 nearby (d ≤ 20 pc) T dwarfs based on forward-modeling analysis of nearly twenty years of high-resolution spectra obtained with Keck/NIRSPEC. We present the distributions of our RV and vsini measurements, and statistically analyze kinematic ages of the local ultracool dwarf population-based on RV and astrometric measurements of 173 late-M, L, and T dwarfs within 20 pc, and show that the L dwarfs as a population are significantly more dispersed and hence older than expected from population simulations. However, the local L dwarf sample has a higher fraction of thick disk sources, and removing them brings the L dwarf age into alignment, resolving a decade-old mystery. A kinematic dispersion break is found around the L4–L6 subtypes, which likely reflects the terminus of the stellar Main Sequence.