In the paper, we study the conditions for the generation of backward runaway electrons through a grounded grid cathode in atmospheric pressure air at high-voltage pulses with a full width at half maximum of 1 ns and risetime of 0.3 ns applied to the gap from a SLEP-150 pulser. The study confirms that backward runaway electrons and X-rays do arise near grid cathodes in atmospheric pressure air. It is shown that the current of the backward beam and the X-rays from the gas diode depend differently on the interelectrode distance. The average X-ray exposure dose in a pulse is more than 3.5 mR.