Concentrated suspensions of hard spheres with a bimodal size distribution provide a system with a very rich behavior, which largely still has to be explored. We present here an experimental study, focused at the kinetics of aggregation between the large spheres, as caused by the presence of the small spheres. At high concentrations of the small particles, the pair potential between the large spheres is considered to contain not only an attractive well in the contact region but also a repulsive barrier at larger separations. This barrier was confirmed to be present in our system and was seen to have a profound influence on the initial aggregation rate. To test the adequacy of the binary hard sphere (BHS) model to describe our system, we have performed quantitative measurements and modeling of the rate of aggregate formation. Three-dimensional video microscopy, with a spatial resolution at the level of single (large) particles, allowed us to determine the time evolutions for monomer, dimer, and trimer number densities. By fitting these to theoretical model expressions, characteristic times were extracted and translated into stability ratios. The experiments were performed with colloidal silica spheres, having respective diameters of 920 and 50 nm for the large and small particles. Particle volume fractions ranged from 0.01 to 0.35% for the large particles and from 30 to 40% for the small particles. Besides experiments under quiescent conditions, we also performed measurements in flow, for Peclet numbers up to 20. The time evolutions of the various aggregation number densities were found to correspond well with Smoluchowski theory. The stability ratio W was found to increase sharply with small-particle concentration, in correspondence with the predictions. The magnitudes of the experimental W were found to be 4-11 (2-3) times larger than those predicted for a BHS in quiescent fluid (under shear).