Transcranial direct current stimulation (tDCS) and transcranial random noise stimulation (tRNS) have been claimed to produce many remarkable enhancements in perception, cognition, learning and numerous clinical conditions. The physiological basis of the claims for tDCS rests on the finding that 1 mA of unilateral anodal stimulation increases cortical excitation and 1 mA of cathodal produces inhibition. Here we show that these classic excitatory and inhibitory effects do not hold for the bilateral stimulation or 2 mA intensity conditions favoured in cognitive enhancement experiments. This is important because many, including some of the most salient claims are based on experiments using 2 mA bilateral stimulation. The claims for tRNS are also based on unilateral stimulation. Here we show that, again the classic excitatory effects of unilateral tRNS do not extend to the bilateral stimulation preferred in enhancement experiments. Further, we show that the effects of unilateral tRNS do not hold when one merely doubles the stimulation duration. We are forced to two conclusions: (i) that even if all the data on TES enhancements are true, the physiological explanations on which the claims are based are at best not established but at worst false, and (ii) that we cannot explain, scientifically at least, how so many experiments can have obtained data consistent with physiological effects that may not exist.