Heterogeneous materials tend to fail at the weakest cross-section, where the presence of microstructural heterogeneities or defects controls the tensile strength. Short-fibre composites are an example of heterogeneous materials, where unwanted fibre agglomerates are likely to initiate tensile failure. In this study, the dimensions and orientation of fibre agglomerates have been analysed from three-dimensional images obtained by X-ray microtomography. The geometry of the specific agglomerate responsible for failure initiation has been identified and correlated with the strength. At the plane of fracture, a defect in the form of a large fibre agglomerate was almost inevitably found. These new experimental findings highlight a problem of some existing strength criteria, which are principally based on a rule of mixture of the strengths of constituent phases, and not on the weakest link. Only a weak correlation was found between stress concentration induced by the critical agglomerate and the strength. A strong correlation was however found between the stress intensity and the strength, which underlines the importance of the size of largest defects in formulation of improved failure criteria for short-fibre composites. The increased use of three-dimensional imaging will facilitate the quantification of dimensions of the critical flaws.