The Japanese atomic bomb survivors that were directly exposed to both γ rays and neutrons have been followed by the Radiation Effects Research Foundation (RERF). The estimation of the γ-ray risks requires some adjustment for the greater biological effect of the neutrons per unit dose. Because the small neutron doses and the predominant γ-ray doses are highly correlated, the neutron relative biological effectiveness (RBE) cannot be reliably estimated from the survivors' data and information from radiobiology must be invoked. As data became available on neutron doses, RERF has used a constant neutron RBE value of 10, even though radiobiological studies indicate that the RBE values appear to have considerably larger values at low doses. The approximation RBE = 10 assumes that if the RBE is variable it takes roughly this value in the range of total dose most relevant for linear risk estimation, namely about 1 Gy. We consider some possible RBE functions to explain the correct use and the impact of a dose-dependent RBE. However, we do not advocate any particular choice or even that a variable RBE be employed. Rather we show that the assumed neutron RBE, within a wide range of choices, is far less important to the outcome of risk assessment of the RERF data than generally believed. Some of these misperceptions have been related to the consideration of variable RBE functions, and without due attention to the fact that in the case of the A-bomb survivors' data, the mixed field of neutrons and γ rays must be considered. Therefore, the RBE value of neutrons is much lower than the RBE in pure neutron fields that are used in radiobiological experiments. Thus, applying the pure neutron field RBE to the mixed-field A-bomb radiation can lead to an overestimation of the actual neutron RBE for moderate total dose levels of 1 Gy by a factor of more than four. While in a pure neutron exposure the RBE depends on the neutron dose, in the mixed field it depends on both components of exposure, and in particular, we show that in the RERF setting the RBE depends mainly on the accompanying γ-ray dose.