  We have investigated the possibility of inferring peculiar velocities for
clusters of galaxies from the Doppler shift of scattered cosmic microwave
background (CMB) photons. We find that if the core radius of the gas
distribution or the beam size of the instrument is larger than 3-7 arcminutes,
then the maximum attainable signal-to-noise ratio is determined by confusion
with primary fluctuations. For smaller angular scales, ``cosmic confusion'' is
less important and instrumental noise and/or foreground emission will be the
limiting factor. For a cluster with the optical depth of the Coma cluster and
for an optimal filtering technique, typical one-sigma errors span the wide
range from 400 to 1600 km/s, depending on the cosmological model, the
resolution of the instrument and the core radius of the cluster. The results
have important implications for the design of future high-resolution surveys of
the CMB. Individual peculiar velocities will be measurable only for a few fast
moving clusters at intermediate redshift unless cosmic fluctuations are smaller
than most standard cosmological scenarios predict. However, a reliable
measurement of bulk velocities of ensembles of X-ray bright clusters will be
possible on very large scales (100-500 Mpc/h).
