  The NMR relaxation rates of (quasi-)one-dimensional spin gapped systems such
as the $S=1/2$ spin ladder with a diagonal interaction and the Haldane-gap
system are investigated at low temperatures in magnetic fields. We show that in
the regime where the gap is collapsed by magnetic fields, the NMR relaxation
rate diverges with decreasing temperature and its exponent shows a behavior
characteristic of the model as a function of magnetic field. We discuss our
results in comparison with recent experimental findings for the spin ladder
system ${\rm Cu_{2}(C_{5}H_{12}N_{2})_{2}Cl_{4}}$ and for the Haldane-gap
system ${\rm (CH_3)_4NNi(NO_2)_3}$.
