The long-time correlation functions of an infinitesimally thin needle moving through stationary point scatterers a so-called Lorentz model exhibits surprisingly long-time tails. These long-time tails are now seen to persist in a two-dimensional model even when the needle has a finite thickness. If the needles are too thick, then the needles are effectively trapped at all nontrivial densities of the scatterers. At needle widths approximately equal to or smaller than sigma = epsilon/20 where epsilon is the average spacing between scatterers, the needle diffuses and exhibits the crossover transition from the expected Enskog behavior to the enhanced translation diffusion seen earlier by Hofling, Frey and Franosch [Phys. Rev. Lett. 2008, 101, 120605]. At this needle width, an increase in its center-of-mass diffusion with respect to increasing density is seen after a crossover density of n* approximate to 5 is reached. (The reduced density n* is defined as n* = nL(2) where n is the number density of particles and L is the needle length.) The crossover transition for needles with finite thickness is spread over a range of densities exhibiting intermediate behavior. The asymptotic divergence of the center of mass diffusion is suppressed compared to that of infinitely thin needles. Finally, a new diminished diffusion regime, apparently due to the increased importance of head-on collisions, now appears at high scattering densities.