  We study smooth threefolds of the projective space of dimension 5 whose
quadrisecant lines don't fill up the space. We give a complete classification
of those threefolds X whose only quadrisecant lines are the lines contained in
X. Then we prove that, if X admits "true" quadrisecant lines, but they don't
fill up the space, then either X is contained in a cubic hypersurface, or it
contains a family of dimension at least two of plane curves of degree at least
four.
