During the last sixty years there have been consistent reports in the literature that in the boundary lubrication regime, some, but not all, solutions of organic friction modifiers give extremely low friction at very low sliding speed, that then increases linearly with the logarithm of sliding speed. This article first reviews some previous studies that show this phenomenon and describes the main mechanisms proposed to explain it. New friction-sliding speed data are then presented, which show that an increase in friction with sliding speed occurs with saturated alkyl chain organic friction modifiers but not with unsaturated chain, oleyl-based ones, at the concentrations studied. It is, however, shown that elaidic acid, the trans-isomer of oleic acid gives friction that increases with sliding speed. A key difference between these two compounds is that the cis arrangement of carbon-carbon bonds around the double bond of oleic acid means that the molecule cannot easily adopt a linear configuration, while elaidic acid can. This suggests that the ability of an organic friction modifier to produce friction that increases with sliding speed originates from its ability to form close-packed layers on steel surfaces. Importantly, even though oleyl derivatives do not show friction that increases with sliding speed, they still reduce friction quite significantly over the sliding speed range studied, although to a lesser extent than their saturated analogues, especially at low sliding speeds.