Using a large N-body cosmological simulation combined with a subgrid treatment of galaxy formation, merging, and tidal destruction, we study the formation and evolution of the galaxy and cluster population in a comoving volume (100 Mpc)(3) in a.CDM universe. At z = 0, our computational volume contains 1788 clusters with mass M-cl > 1.1x10(12) M-circle dot, including 18 massive clusters with M-cl > 10(14) M-circle dot. It also contains 1,088,797 galaxies with mass M-gal >= 2 x 10(9) M-circle dot and luminosity L > 9.5 x 10(5) L-circle dot. For each cluster, we identified the brightest cluster galaxy (BCG). We then computed two separate statistics: the fraction f(BNC) of clusters in which the BCG is not the closest galaxy to the center of the cluster in projection, and the ratio Delta upsilon/sigma, where Delta upsilon is the difference in radial velocity between the BCG and the whole cluster and sigma is the radial velocity dispersion of the cluster. We found that f(BNC) increases from 0.05 for low-mass clusters (M-cl similar to 10(12) M-circle dot) to 0.5 for high-mass clusters (M-cl > 10(14) M circle dot) with very little dependence on cluster redshift. Most of this result turns out to be a projection effect and when we consider three-dimensional distances instead of projected distances, f(BNC) increases only to 0.2 at high-cluster mass. The values of Delta upsilon/sigma vary from 0 to 1.8, with median values in the range 0.03-0.15 when considering all clusters, and 0.12-0.31 when considering only massive clusters. These results are consistent with previous observational studies and indicate that the central galaxy paradigm, which states that the BCG should be at rest at the center of the cluster, is usually valid, but exceptions are too common to be ignored. We built merger trees for the 18 most massive clusters in the simulation. Analysis of these trees reveal that 16 of these clusters have experienced 1 or several major or semi-major mergers in the past. These mergers leave each cluster in a non-equilibrium state, but eventually the cluster settles into an equilibrium configuration, unless it is disturbed by another major or semi-major merger. We found evidence that these mergers are responsible for the off-center positions and peculiar velocities of some BCGs. Our results thus support the merging-group scenario, in which some clusters form by the merging of smaller groups in which the galaxies have already formed, including the galaxy destined to become the BCG. Finally, we argue that f(BNC) is not a very robust statistics, as it is very sensitive to projection and selection effects, but that Delta upsilon/sigma is more robust. Still, both statistics exhibit a signature of major mergers between clusters of galaxies.