The compressive response of a three-dimensional (3D) non-interlaced composite comprising three orthogonal sets of carbon fibre tows within an epoxy matrix is analysed. First, the compressive response is measured in three orthogonal directions and the deformation/failure modes analysed by a combination of X-ray tomography and optical microscopy. In contrast to traditional unidirectional and two-dimensional (2D) composites, stable and multiple kinks (some of which zig-zag) form in the tows that are aligned with the compression direction. This results in an overall composite compressive ductility of about 10% for compression in the low fibre volume fraction direction. While the stress for the formation of the first kink is well predicted by a usual micro-buckling analysis, the composite displays a subsequent hardening response associated with formation of multiple kinks. Finite element (FE) calculations are also reported to analyse the compressive response with the individual tows modelled as anisotropic continua via a Hill plasticity model. The FE calculations are in good agreement with the measurements including prediction of multiple kinks that reflect from the surfaces of the tows. The FE calculations demonstrate that the three-dimensionality of the microstructure constrains the kinks and this results in the stable compressive response. In fact, the hardening and peak strength of these composites is not set by the tows in direction of compression, but rather set by the out-of-plane compressive response of the tows perpendicular to the compression direction.