We present a numerical study of two-particle dispersion from point sources in three-dimensional incompressible homogeneous and isotropic turbulence at Reynolds number Re¿300. Tracer particles are emitted in bunches from localized sources smaller than the Kolmogorov scale. We report the first quantitative evidence, supported by an unprecedented statistics, of the deviations of relative dispersion from Richardson’s picture. Deviations are due to extreme events of pairs separating much faster than average, and of pairs remaining close for long time. The two classes of events are the fingerprints of complete different physics, the former dominated by inertial subrange and large-scale fluctuations, and the latter by dissipation subrange. A comparison of the relative separation in surrogate white-in-time velocity field, with correct viscous-, inertial-, and integral-scale properties, allows us to assess the importance of temporal correlations along tracer trajectories.