Carbon nanotube (CNT) fibres, especially if perfect in terms of purity and alignment, are of extreme anisotropy. With their high axial strength but ready slippage between the CNTs, there is utmost difficulty in transferring the force applied uniformly. Finite element analysis is used to predict the stress distribution in CNT fibres loaded by grips attached to their surface, along with the resulting tensile stress-strain curves. This study demonstrates that in accordance with St Venant’s principle very considerable length-to-diameter ratios (~ 103) are required before the stress becomes uniform across the fibre, even at low strains. It is proposed that lack of perfect orientation and presence of carbonaceous material between bundles greatly enhances the stress transfer, thus increasing the load it can carry before failing by shear. It is suggested that a very high strength batch of fibres previously observed experimentally had an unusually high concentration of internal particles, meaning that the pressure exerted by the grips would assist stress transfer between the layers. We conclude, that the strength of CNT fibres depends on the specific testing geometries and that imperfections, whether by virtue of less-than-perfect orientation or of embedded impurities, are actually major positive contributors to the observed strength.