The existence of fire spread limits along low flammable metal pieces and the parameters affecting them have been studied theoretically. The governing equations for the upward fire spread along a metal rod have been solved by considering the heat release rate in the molten mass and the heat transfer from the burning molten mass to the unburned solid metal and to the environment. The numerical calculations have been performed for a copper rod. The burning temperature and the fire spread rate have been obtained as functions of the oxygen pressure, the metal temperature, and the ambient temperature. The relations between the burning temperature and these parameters are expressed by S-shaped curves. The critical values of the above parameters, below which the fire cannot spread, are obtained as the values at the nose points of the S-shaped curves. They show that fire spread limits exist corresponding to the metal temperature, the ambient temperature, and the oxygen pressure. The critical metal temperature below which the fire cannot spread decreases with the increase of the oxygen pressure and the ambient temperature. The critical oxygen pressure, below which the fire cannot spread, decreases with the increase of the metal temperature and the ambient temperature. This analysis shows that the high-pressure tests of fire spread for the low flammable metals are indispensable to prevent fire accidents in high-pressure oxygen systems.