The industrially produced chemical pulps have lower strength properties than those obtained under laboratory conditions, and this difference is referred to as the strength delivery (SD) problem. In this study, the hypothesis was put forward that the SD could, at least in part, be accounted for by the supramolecular structure of the cellulose microfibrils of the fiber wall. To test the hypothesis, two bleached softwood kraft pulps (BSKP) were manufactured from the same starting material with different degrees of cellulose aggregation, but the pulps were otherwise as similar as possible in other controllable respects. The chemical and physical properties, including the pulp strength, were tested. A selective increase of the degree of cellulose microfibril aggregation resulted in a pulp with a decreased tear index (TI) at a specified tensile index, and this decrease was similar in magnitude to what is typically encountered in SD. Accordingly, the current experimental study succeeded in mimicking the SD problem. The lateral fibril aggregate dimensions (LFAD) seem to play a pivotal role and it can be safely concluded in general that the supramolecular structure of cellulose in the fibers may be an important factor contributing to the SD problem.