  We explore some of the effects that discreteness and two-body scattering may
have on N-body simulations with ``realistic'' cosmological initial conditions.
We use an identical subset of particles from the initial conditions for a
$128^3$ Particle-Mesh (PM) calculation as the initial conditions for a variety
P$^3$M and Tree code runs. We investigate the effect of mass resolution (the
mean interparticle separation) since most ``high resolution'' codes only have
high resolution in gravitational force. The phase-insensitive two--point
statistics, such as the power spectrum (autocorrelation) are somewhat affected
by these variations, but phase-sensitive statistics show greater differences.
Results converge at the mean interparticle separation scale of the lowest
mass-resolution code. As more particles are added, but the force resolution is
held constant, the P$^3$M and the Tree runs agree more and more strongly with
each other and with the PM run which had the same initial conditions. This
shows high particle density is necessary for correct time evolution, since many
different results cannot all be correct. However, they do not so converge to a
PM run which continued the fluctuations to small scales. Our results show that
ignoring them is a major source of error on comoving scales of the missing
wavelengths. This can be resolved by putting in a high particle density. Since
the codes never agree well on scales below the mean comoving interparticle
separation, we find little justification for quantitative predictions on this
scale. Some measures vary by 50%, but others can be off by a factor of three or
more. Our results suggest possible problems with the density of galaxy halos,
formation of early generation objects such as QSO absorber clouds, etc.
