In two fast neutron data sets, comprising in vitro and in vivo experiments, an inverse relationship is found between the low-linear energy transfer (LET) α/β ratio and the maximum value of relative biological effect (RBEmax), while the minimum relative biological effect (RBEmin) is linearly related to the square root of the low-LET α/β ratio. RBEmax is the RBE at near zero dose and can be represented by the ratio of the α parameters at high- and low-LET radiation exposures. RBEmin is the RBE at very high dose and can be represented by the ratio of the square roots of the β parameters at high- and low-LET radiation exposures. In principle, it may be possible to use the low-LET α/β ratio to predict RBEmax and RBEmin, providing that other LET-related parameters, which reflect intercept and slopes of these relationships, are used. These two limits of RBE determine the intermediate values of RBE at any dose per fraction; therefore, it is possible to find the RBE at any dose per fraction. Although these results are obtained from fast neutron experiments, there are implications for charged particle therapy using protons (when RBE is scaled downwards) and for heavier ion beams (where the magnitude of RBE is similar to that for fast neutrons). In the case of fast neutrons, late reacting normal tissue systems and very slow growing tumours, which have the smallest values of the low-LET α/β ratio, are predicted to have the highest RBE values at low fractional doses, but the lowest values of RBE at higher doses when they are compared with early reacting tissues and fast growing tumour systems that have the largest low-LET α/β ratios.