  An important issue in the direct reconstruction method of determining the $W$
mass from $q\bar q Q\bar Q$ events at LEP2 concerns the impact of the
relatively unknown QCD interconnection effects. It has been suggested that a
study of `short string' states, in which colour singlet states are formed from
$q \bar Q $ and $Q\bar q$ pairs with small phase--space separation, could shed
important light on this issue. We show that such configurations can also be
generated by conventional background $e^+e^-\ar 4$~parton processes, in
particular QCD $q \bar q g g$ and $q \bar q Q \bar Q$ and non--resonant
electroweak $q \bar q Q \bar Q$ production. We study the colour and kinematic
structure of these background contributions, and estimate the event rate to be
expected at LEP2. We find that the QCD processes are heavily suppressed, but
that non--resonant $q \bar q Q \bar Q$ production may be comparable in rate to
the expected `short string' signal from $W^+W^-$ production.
