We consider the problem of achieving reliable communication with quiescent algorithms (i.e., algorithms that eventually stop sending messages) in asynchronous systems with process crashes and lossy links, and show that, among failure detectors with bounded output size, less than or greater than P is the weakest one that can be used to solve this problem. Combined with a result in [ACT97a], this shows that failure detectors that are commonly used in practice, i.e., those that output lists of suspects, are not always the best ones to solve a problem.