  Bose-Einstein condensates are studied in a potential of finite depth which
supports both bound and quasi-bound states. This potential, which is harmonic
for small radii and decays as a Gaussian for large radii, models experimentally
relevant optical traps. The nonlinearity, which is proportional to both the
number of atoms and the interaction strength, can transform bound states into
quasi-bound ones. The latter have a finite lifetime due to tunnelling through
the barriers at the borders of the well. We predict the lifetime and stability
properties for repulsive and attractive condensates in one, two, and three
dimensions, for both the ground state and excited soliton and vortex states. We
show, via a combination of the variational and WKB approximations, that
macroscopic quantum tunnelling in such systems can be observed on time scales
of 10 milliseconds to 10 seconds.
