Google began testing autonomous vehicles in California in 2009. Though that wasn't technically illegal (because lawmakers in the early 20th century never saw the need to require human drivers in cars, since they assumed one would always be onboard), California passed legislation in 2012 that officially grants Google and others permission to test self-driving cars on the state's roads.
Two years later, California's Department of Motor Vehicles caught up with the new law and issued regulations related to autonomous vehicle testing. One of those regulations requires companies to reveal how many accidents their self-driving cars have had. According to the Boston Herald, Google's autonomous vehicles have had a total of three accidents since regulations went into effect last September, and just eight others in the previous five years (when, to be fair, much of the testing was done on tracks or closed roads).
Google says that all of the accidents have been minor, with no associated injuries. It also says that each fender-bender was caused by human drivers.
In other words, the autonomous car itself was never at fault.
Google hasn't provided additional details, and as the Herald notes, it's impossible to verify whether the company is telling the truth. But if so, it's not a bad record for a fleet of 23 test vehicles that's racked up an impressive 1.7 million miles on the odometer, roughly 1 million of which have occurred in self-driving mode. It also jibes with previous reports that have suggested that Google's cars have only been involved in accidents when human drivers have been in control of the vehicles or when another car has been at fault.
Could Google be fibbing? Possibly. If a self-driving car were to have major accident while in autonomous mode, it would expose the company to a very major lawsuit, since California law would likely consider the autonomous technology to be a design flaw. It would also create headlines that could delay the rollout of autonomous features and cars to the public. That, in turn, would take a huge bite out of Google's investment (and Apple's, and Tesla's, and every other automaker working on self-driving technology).
On the other hand, in the Facebook Age, it would be pretty hard to keep a major accident entirely hushed up. If Google tried and were exposed, it could turn into a PR fiasco. More likely than not, Google is being entirely (or mostly) forthright.
In other words, the answer is still no, you can't drive better than an autonomous car. For now, anyway.