  Spin-lattice relaxation rate $T_1^{-1}$ of $^1$H-NMR has been measured in
(CH$_3$)$_2$CHNH$_3$Cu(Cl$_x$Br$_{1-x}$)$_3$ with $x=0.88$, which has been
reported to be gapped system with singlet ground state from the previous
macroscopic magnetization and specific heat measurements, in order to
investigate the bond randomness effect microscopically in the gapped composite
Haldane system (CH$_3$)$_2$CHNH$_3$CuCl$_3$. It was found that the spin-lattice
relaxation rate $T_1^{-1}$ in the present system includes both fast and slow
relaxation parts indicative of the gapless magnetic ground state and the gapped
singlet ground state, respectively. We discuss the obtained results with the
previous macroscopic magnetization and specific heat measurements together with
the microscopic $\mu$SR experiments.
