The role of the nuclear and the Coulomb interactions in the dissociati
on of the neutron-rich exotic nuclei Li-11, Be-11 and Be-14 incident o
n a number of targets has been studied. The two-neutron removal cross
sections of Li-11 and Be-14 and one-neutron removal cross section of B
e-11 have been calculated. The theory of diffraction dissociation and
diffraction stripping has been used to account for the nuclear effects
. The Coulomb-dissociation cross sections have been calculated within
the framework of a direct fragmentation model formulated earlier. The
Coulomb dissociation along with the nuclear cross sections explain sat
isfactorily the target-mass dependence of the experimental total two-n
eutron removal cross sections of Li-11 and Be-14 around 30 MeV/A incid
ent energy. The beam-energy dependence of this cross section is also w
ell reproduced. The stripping contributions dominate the nuclear cross
sections. The dependence of the reaction mechanism underlying the two
-neutron removal process on the separation energy of the two-neutron s
ystem in the projectile is discussed. The role of the ''post-accelerat
ion effect'' on the energy spectra of the produced fragments is invest
igated.