Carbon nanotube (CNT) fibres, especially if perfect in terms of their purity and alignment, are extremely anisotropic. With their high axial strength but ready slippage between the CNTs, there is utmost difficulty in transferring uniformly any applied force. 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 the fibre 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, can act as major positive contributors to the observed strength.