  It has recently been argued that non--BPS brane world scenarios can reproduce
the small value of the cosmological constant that seems to have been measured.
Objections against this proposal are discussed and necessary (but not
sufficient) conditions are stated under which it may work. At least n=2 extra
dimensions are needed. Also, the mass matrix in the supergravity sector must
satisfy $Str M^2=0$. Moreover, the proposal can be ruled out experimentally if
Newton's constant remains unchanged down to scales of 10 micrometers. If, on
the other hand, such a ``running Newton constant'' is observed, it could
provide crucial experimental input for superstring phenomenology.
