The Purcell effect is usually described as a modification of the spontaneous decay rate in the presence of a resonator. In plasmonics, this effect is commonly associated with a large local-field enhancement in "hot spots" due to the excitation of surface plasmons. However, high-index dielectric nanostructures, which become the basis of all-dielectric nanophotonics, cannot provide high values of the local-field enhancement due to larger radiation losses. Here, we demonstrate how to achieve a strong Purcell effect in all-dielectric nanostructures, and show theoretically that the Purcell factor can be increased by two orders of magnitude in a finite chain of silicon nanoparticles. Using the eigenmode analysis for an infinite chain, we demonstrate that the high Purcell factor regime is associated with a Van Hove singularity. We perform a proof-of-concept experiment for microwave frequencies and observe the 65-fold enhancement of the Purcell factor in a chain of 10 dielectric particles.