We show that spatiotemporal light bullets generated by self-focusing and filamentation of 100 fs, 1.8 μm pulses in a dielectric medium with anomalous group velocity dispersion (sapphire) are extremely robust to external perturbations. We present the experimental results supported by the numerical simulations that demonstrate complete spatiotemporal self-reconstruction of the light bullet after hitting an obstacle, which blocks its intense core carrying the self-compressed pulse, in nonlinear as well as in linear (free-space) propagation regimes.