  We analyze the uncertainties and possible systematics associated with the
"Dark Flow" measurements using the cumulative Sunyaev-Zeldovich effect combined
with all-sky catalogs of clusters of galaxies. Filtering of all-sky cosmic
microwave background (CMB) maps is required to remove the intrinsic
cosmological signal down to the limit imposed by cosmic variance. Contributions
to the errors come from the remaining cosmological signal, that integrates down
with the number of clusters, and the instrumental noise, that scales with the
number of pixels; the latter decreases with integration time and is subdominant
for the Wilkinson Microwave Anisotropy Probe 5-year data. It is proven both
analytically and numerically that the errors for 5-year WMAP data are $\simeq
15\sqrt{3/N_{\rm clusters}} \mu$K per dipole component. The measured bulk flow
velocity is significant at up to the $>99.95%$ confidence level, depending on
the cluster sample. We discuss different methods to compute error bars and
demonstrate that they have biases that would overpredict the errors, as is the
case in a recent reanalysis of our earlier results. If the signal is caused by
systematic effects present in the data, such systematics must have a dipole
pattern, which correlates with cluster X-ray luminosity and is present only at
cluster positions. Only contributions from the Sunyaev-Zeldovich effect could
provide such contaminants via several potential effects. We discuss such
candidates apart from the bulk-motion of the cluster samples and demonstrate
that their contributions to our measurements are negligible. (abridged)
