We consider the dynamics of non-interacting Brownian particles which are driven by correlated (non-independent) noise sources. In simple confining potentials the particles tend to aggregate as the noise correlation is increased. If two particles are subject to the same noise they will coalesce and remain together ever after. We show that complete aggregation of the particles can be expected even in the case of a disordered potential which does not confine the individual particle trajectories. Finally, we examine the case of correlation in the noises which depends on the separation of the two particles.