  We have calculated the leading color-octet contributions to the production of
$J/\psi$ particles in photon-proton collisions. Using the values for the
color-octet matrix elements extracted from fits to prompt $J/\psi$ data at the
Tevatron, we demonstrate that distinctive color-octet signatures should be
visible in $J/\psi$ photoproduction. However, these predictions appear at
variance with recent experimental data obtained at HERA, indicating that the
phenomenological importance of the color-octet contributions is smaller than
expected from theoretical considerations and suggested by the Tevatron fits.
