Measurements of $^{119}\mathrm{Sn}$ nuclear spin-lattice relaxation rate, 1/${\mathit{T}}_{1}$, and Knight shift, K, in the off-stoichiometric compound ${\mathrm{CeNi}}_{1.01}$Sn and the substituted compounds ${\mathrm{CeNi}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathit{M}}_{\mathit{x}}$Sn (M=Cu,Co) and ${\mathrm{Ce}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{La}}_{\mathit{x}}$NiSn have been made in order to unravel the impurity and/or carrier doping effects on the V-shaped gapped state in CeNiSn. From the detailed analysis of the T dependence of 1/${\mathit{T}}_{1}$, it is shown that the density of states (DOS) is induced just at the Fermi level for ${\mathrm{CeNi}}_{1.01}$Sn, whereas the DOS increases progressively with the dopant in a finite-energy range near the Fermi level for the substituted compounds. The respective substitution of Co and La into Ni and Ce sites changes the gapped state into the nonmagnetic Fermi-liquid state, whereas the replacement of Ni by Cu with x\ensuremath{\gtrsim} 0.06 gives rise to the antiferromagnetic (AF) ground state. It is suggested that the AF order is realized by the combined effect of the rapid collapse of the V-shaped gap and the increase of the DOS at the Fermi level. It has been found to lead to the magnetic ground state and the nonmagnetic Fermi-liquid state to dope electrons and holes into the renormalized conduction bands, respectively. \textcopyright{} 1996 The American Physical Society.