Sadly this does happen all the time. Usually people panic and press harder on the pedal, still convinced in their mind that "I'm pressing the brake not the gas". Modern cars can accelerate quite quickly from a stop.
http://www.caranddriver.com/feature...toyotas-unintended-acceleration-scare-feature
This. There's been some behavioral research done on this and, from what I understand, it's like a mental short circuit. You think you're hitting the breaks but you're not, and the fact that you're not shopping just makes you jam down the pedal you think is the break even harder. There have even been deadly accidents.
Some recommend that you should teach people in driving school to let go of all pedals if it feels like the car is accelerating out of control and just steer. That way you'd at least stop accelerating and eventually stop completely, especially since there have been cases where people keep accelerating for minutes without realising they're pressing the wrong pedal down.