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A large set of rising adiabatic parcel simulations is executed to investigate the combined diffusional and accretional growth of cloud droplets in maritime and continental conditions, and to assess the impact of enhanced droplet collisions due to small-scale cloud turbulence. The microphysical model applies the droplet number density function to represent spectral evolution of cloud and rain/drizzle drops, and various numbers of bins in the numerical implementation, ranging from 40 to 320. Simulations are performed applying two traditional gravitational collection kernels and two kernels representing collisions of cloud droplets in the turbulent environment, with turbulent kinetic energy dissipation rates of 100 and 400 cm<sup>2</sup> s<sup>−3</sup>. The overall result is that the rain initiation time significantly depends on the number of bins used, with earlier initiation of rain when the number of bins is low. This is explained as a combination of the increase of the width of activated droplet spectrum and enhanced numerical spreading of the spectrum during diffusional and collisional growth when the number of model bins is low. Simulations applying around 300 bins seem to produce rain at times which no longer depend on the number of bins, but the activation spectra are unrealistically narrow. These results call for an improved representation of droplet activation in numerical models of the type used in this study. <br><br> Despite the numerical effects that impact the rain initiation time in different simulations, the turbulent speedup factor, the ratio of the rain initiation time for the turbulent collection kernel and the corresponding time for the gravitational kernel, is approximately independent of aerosol characteristics, parcel vertical velocity, and the number of bins used in the numerical model. The turbulent speedup factor is in the range 0.75–0.85 and 0.60–0.75 for the turbulent kinetic energy dissipation rates of 100 and 400 cm<sup>2</sup> s<sup>−3</sup>, respectively.