Actually the original 7.62 Nato was designed, at the request of the US military, to chamber and "wedge" headspace in the 30 caliber cartridge of 1906 chamber. They had quite a stockpile of 30-06 rifles and wanted to be able to fire the new cartridge safely from an 06. If you notice the shoulder dia of the 308/7.62 it is slightly larger than the corresponding dia at the same body length on the 30-06. This does assume though that chambers are cut to the proper dimensions and factories make their cartridges to proper dimensions, both of which we all know to have significant variances.
In most cases though you can safely and even somewhat accurately fire 308 in '06 chambers..............were I in a war zone scenario and had my trusty '06 and only 7.62 ammo was available, you can be assured my rifle would not be silent.