Nanocrystalline (n-) LiNbO3 samples with average grain sizes between 16 and 105 nm were prepared from polycrystalline (p-) material with an average grain size of the order of one micrometer by high-energy ball milling. NMR investigations of (i) the Li-7 spin-lattice relaxation (SLR) rate T-1(-1) in the laboratory and T-1e(-1) in the pulsed rotating reference frame and of (ii) Li-7 spectra, in particular line shapes and motional narrowing (MN) of the central line, were performed in the temperature range from 300 K to a maximum of 1400 K in the case of p-LiNbO3 and from 140 to 460 K in the case of n-LiNbO3. The following results were obtained. (1) The SLR rate measurements yield an apparent activation energy of the Li diffusion in n-LiNbO3 that is about 1/3 of the value obtained for the p-material. (2) The frequency dependence of the SLR rate according to T-1(e)(-1) proportional to v(-beta) with beta in the range from 1. 1 to 1.5 as well as the asymmetry of the diffusion-induced peak in the log T-1e(-1) vs T-1 diagram of p-LiNbO3 are proving non-BPP behavior for both samples. (3) In n-LiNbO3 MN starts already at 250 K, i.e., about 400 K lower than in p-LiNbO3, and reflects an apparent activation energy that is approximately 1/3 of the value found for the p-material. (4) In contrast to p-LiNbO3, with increasing temperature the Li-7 NMR spectra of n-LiNbO3 are revealing a characteristic structure of the central line, namely a superposition of two contributions. This is regarded as a consequence of the different dynamic properties of atoms in the interfacial regions (IR) and in the grains. From the spectrum at 450 K the fraction of atoms belonging to IR can be estimated. (5) The intensities of the quadrupole satellites showing different temperature dependencies in the p- and n-samples are indicating some exchange between the two spin reservoirs 'IR' and 'grains'. This leads to the hypothesis that n-ceramics cannot simply be regarded as heterogeneous materials where the two types of zones, i.e., IR and grains, are independent and closed.