Electric vehicles require large quantities of electricity, so lots of electric vehicles would strain the power grid. If a few electric vehicles were suddenly charging, the increase in demand for electricity could cause the surrounding area to experience electrical shortages. As a result, if everyone had an electric vehicle, there would not be enough power to charge all of those cars. In order to combat this problem, the infrastructure of the electrical grid needs to be updated. This would require massive infrastructure changes, so it is unlikely to happen on a large scale. Most power grids are set up in a way that accounts for the lower electricity use at night while people are asleep. Electric vehicles would conflict with this schedule. Most people would charge them at night--straining the grid when it is at it's lowest.
There is enough electricity in the system to charge a large number of electric vehicles. It is possible that if everyone were to plug in their cars at the same time, during peak hours of electrical use, there could be a shortage. However, that is unlikely to happen. People who buy electric vehicles can be informed about when they ought to charge their cars, and some states have started point systems to encourage people to charge their cars at specific times.