Understanding warm rain initiation through droplet collision and coalescence is a fundamental yet complex challenge in cloud microphysics. Although it is well-known that sufficiently large droplets, so-called precipitation embryos (PEs), may accelerate droplet collisions, it is uncertain how many and how large these PEs should be to affect rain initiation substantially. We address this question using an ensemble of box simulations with Lagrangian cloud microphysics. We found that the warm rain initiation is substantially accelerated only if the PE size or number (or the product of those) exceeds a critical threshold necessary to compensate for the PE-induced suppression of collisions among non-PEs. The sensitivity of this threshold to the shape of the droplet size distribution and turbulence effects on the collision process is analyzed. It is shown that more and larger PEs are needed when collisions are already efficient without PEs. Beyond increasing our fundamental understanding of the precipitation process in warm clouds, our results may help to constrain the effect of PE-like particles intentionally or unintentionally added in geoengineering approaches, such as rain enhancement or marine cloud brightening.