Meh. That's kind of wodgy. The reason that Civ starts you right at the dawn of agriculture is that agriculture (and thus a stable, storable food supply) is what allowed permanent settlements to exist.
Actually 3,000 years before the development of cities there were small scale settlements, inhabited year round, in existance in the middle east, where the first cities would emerge of course. They existed around temples, which would have there food supplied from without in exchange for religious services, trade hubs near a praticularly valuable resource, who would have traded their profits for food. Nevalı Çori and Göbekli Tepe are both dated before the occurance of agrculture, Nevali Cori is a settlement around a temple and Gobekli Tepe is a medium sized temple, on the scale of societies having just inveneted agriculure, but both were built before the discovery of agriculture.
In all likely hood the first cities were places of religious significance, that would attract worshipers from miles around in order to pray at the temple, and in turn feed the priests and towns people. Hence they would take advantage of the multiple hunter-gather bands around their location to provide them with food.
So to tie this all back into civilization, I think it is perfectly reasonable to allow small size cities up unitl agriculture, which is all you can usually manage to create before you discover agriculture in Cavemen to Cosmos. Oh and food storage and preservation techniques existed before agriculture so the hunting bands wouldn't have to visist year round.