If you constructed a wavepacket for the finite well representing a bound particle i.e. one with <E> < 0, would you ever measure it with E > 0? The finite well we are discussing has a potential 0 except between -a and a where it is -V_0. I would say yes because there is nonzero probability that the particle could be found outside the well i.e. |psi(x)|^2 is nonzero for x > |a|. But I am not sure whether that implies that the energy is greater than 0 as well?