In the past I was also explicitly told that the slowest reaction is "often" the rate determining step, and like you I figured some day I would find a reaction where the fastest step determines the rate. Thinking over it a bit more now, however, I believe we may have misinterpreted what was meant. Rather than looking for an opposite situation, it is likely we were indirectly told that in some cases the rate-determining-step picture simply isn't applicable in the first place.
In general, reactions can't be modelled though such simple kinetic theory; in truth, when a bunch of reactants are brought together, all steps in every possible reaction route matter. It just happens that for several simple but relevant chemical systems there are few possible steps and few accessible reaction routes, and they have such wildly different rate constants that we can approximate by looking only at the slowest step of the fastest route. A more complex reaction with several steps, several side-reactions and similar rate constants will likely be poorly described by a RDS framework, so one can say that the reaction rate is not determined by the slowest step because the reaction rate won't be well determined by any single step.