Theoretically, yes, if enough electric vehicles were charged in the same time during peak times in a local area. But frankly this situation will not happen in real life because we’re a long way from that. The process of penetration of electric-car penetration and smart grid technology are upgrading thank to the concentration of public in electric vehicles.
According to the studies of Idaho and Pacific Northwest National Labs, the electric grid power of United State has ability to support more than 1 million electric vehicles at off-peak times, without applying any system’s upgrade.
The risk is also reduced by the fact that people will charge their car at night, when the power demand of grid is lower.
Electric car producer – especially Tesla motors – promise to build charging facility. Tesla will build their supercharging network using solar energy. These facility will allow you to charge your car freely, using energy from Sunlight, without affect the working of national electric grid system.