Li-7 nuclear magnetic resonance spin-lattice relaxation time T-1 versus temperature is reported in the 150 K-900 K temperature range on lithium lanthanum titanate fast ionic conductors. Because of the presence of disorder in the distribution of the lanthanum ions in the crystalline structure of this oxide and consequently in the conduction pathways of the lithium ions we propose to explain the strong asymmetry shown by these T-1 versus 1/T curves by assuming independent ionic hops over a distribution of activation energies for the thermally activated Li+ ion hops. According to this assumed model the spin-lattice relaxation times T-1 and the DC conductivity are fitted consistently in the 200-600 K and 300-400 K temperature ranges respectively. For both lower and higher temperatures a departure of the experimental data from the model is observed and explained. The use of this model to fit both T-1 and DC conductivity data ruled out the possibility that different forms of the distribution would lead to a reasonable representation of T-1. The physical meaning of the obtained parameters is discussed in accordance with the structure of the compounds.