The most likely scenario seems to be that Paleolithic humans may have gathered grass seeds ("grains") periodically, when they were available, and ate them if they were hungry enough. This is a long way from depending on them as a food source, or even making up a significant fraction of their diet.
Second, and most importantly, non-domesticated grains have a very short harvest season. Therefore, even if we claim that Paleolithic people subsisted entirely on grains during the time they were available (a claim for which there is no evidence, and plenty to the contrary) it would have been impossible to do so for more than a week or two in the fall.
JS