The inclusion of anisotropy in P-wave traveltime tomography has been undertaken by several authors and in all cases some loss of resolution and uniqueness compared to the isotropic problem was observed. The origin of this problem is analysed for straight-ray tomography using the Radon transform and the projection slice theorem. This analysis shows that the separation of the anisotropy from the isotropic velocity field can only be guaranteed for the dc component. Resolution at higher spatial frequencies depends upon the spatial support of the object, that is the separation of the holes in crosshole work, and the range of ray angles available. Specific calculations in the crosshole case suggest that even in media known to be horizontally stratified, it will probably be difficult to estimate spatially varying elliptical anisotropy at wavelengths less than a few times smaller than the hole separation, and Thomsen's parameter, delta at wave-lengths less than the hole separation itself. These results broadly agree with the empirical observations in the literature.