A new mechanism of neutron acceleration is discussed and studied experimentally in detail for cold neutrons passing through the accelerated perfect crystal with the energies close to the Bragg one. The effect arises due to the following reason. The crystal refraction index (neutron-crystal interaction potential) for neutron in the vicinity of the Bragg resonance sharply depends on the parameter of deviation from the exact Bragg condition, i.e. on the crystal-neutron relative velocity. Therefore the neutrons enter into accelerated crystal with one neutron-crystal interaction potential and exit with the other. Neutron kinetic energy cannot vary inside the crystal due to its homogeneity. So after passage through such a crystal neutrons will be accelerated or decelerated because of the different energy change at the entrance and exit crystal boundaries.