Magnetic resonance studies of nuclear spins in solids are exceptionally well suited to probe the limits of statistical physics. We report experimental results indicating that isolated macroscopic systems of interacting nuclear spins possess the following fundamental property: spin decays that start from different initial configurations quickly evolve towards the same long-time behavior. This long-time behavior is characterized by the shortest ballistic microscopic time scale of the system and therefore falls outside of the validity range for conventional approximations of statistical physics. We find that the nuclear free-induction decay and different solid echoes in hyperpolarized solid xenon all exhibit sinusoidally modulated exponential long-time behavior characterized by identical time constants. This universality was previously predicted on the basis of analogy with resonances in classical chaotic systems.