The paper considers statistical properties of ensembles of chain conformations obtained by short-time Brownian dynamics (BD) of a coarse-grained DNA model in order to find out if the conditions necessary for accurate evaluation of the polymer elasticity are attainable in atom-level molecular dynamics (MD) simulations. To measure the bending persistence length (PL) with a 10% error using data accumulated in a single trajectory of a double helix of 15 base pairs, dynamics should be continued for a few microseconds. However, these estimates should be scaled down by about 2 orders of magnitude because the bending dynamics of short double helices in MD features much smaller relaxation times. As a result, good qualitative agreement with the worm-like chain (WLC) theory is reached in MD after tens of nanoseconds. The presently accessible durations of MD trajectories provide reasonably accurate evaluation of DNA elasticity and allow modeling of its mesoscopic properties. The surprisingly fast bending dynamics of short double helices in MD suggests that the microscopic mechanisms of DNA flexibility differ from a simple harmonic model.