Transverse magnetization of spins diffusing in a bounded region in the presence of a constant field gradient is studied. We investigate the breakdown at short times of the much used formula for the Hahn echo amplitude in a constant gradient in unbounded space : M(2tau)/M(0) = exp(-2D0g2tau3/3). Here D0 is the diffusion constant in unbounded space and g is the field gradient multiplied by the gyromagnetic ratio. We find that this formula is replaced by M(2tau)/M(0) = exp[-2D(eff)g2tau3/3 + O(D0(5/2)g4tau13/2S/V)] with an effective diffusion coefficient D(eff)(2tau) = D0[1-alpha square-root D0tau(S/V) + ...], where alpha is a constant and S/V is the surface to volume ratio of the bounded region. Breakdown is complex but we find that the interplay between a natural length scale l(c) = (g/D0)-1/3 and the geometry of the region governs the problem. The long-time behavior of the free induction decay and echo amplitude are then considered where pure exp[-const t] decay is expected. We consider some simple geometries and find in addition to the well-known result, InM(z,t) approximately -D0g2R(p)4t, valid for R(p) much less than l(c) (where R(p) is the size of the confining space) that in the regime R(p) much greater than l(c) the decay becomes lnM(z,t) approximately -g2/3D0(1/3) t. We then argue that this latter result should apply to more general geometries. We discuss implications for realistic experimental echo measurements and conclude that the g2/3D(0)1/3 decay regime is hard to measure. Implications for the effect of edge enhancement in NMR microscopy are also discussed.