This study investigates the experimental conditions necessary to observe a non-monotonic dependence of shear stress on quasi-static shear strain in structured magnetorheological (MR) fluids of a known composition. The research varies several key parameters, including the concentration and size of magnetic particles, the working gap between rheometer plates, the strength of the external magnetic field, and the material properties of the plates. Rheometric measurements are complemented by microstructural observations of low-concentration MR fluid specimens using X-ray computed microtomography. The results demonstrate that non-monotonic shear stress-strain behavior is characteristic of diluted MR suspensions with linear chain-like particle aggregates. Crucially, this phenomenon occurs only when using rheometer plates with magnetic microparticles embedded in their surface, and the microparticles must be of a size comparable to or larger than the transverse dimensions of the chain aggregates. Conversely, monotonic shear stress increases are observed with non-magnetic or entirely magnetic plates, irrespective of the MR fluid composition. These findings highlight the critical role of particle-structure interactions with the measuring geometry and provide insights into operating conditions for applications requiring precise control of static yield stress. The results also offer valuable experimental data to refine and validate models of MR fluid behavior under quasi-static shear.