Let's get to what Obama actually said, in context:

"If you were successful, somebody along the line gave you some help. There was a great teacher somewhere in your life. Somebody helped to create this unbelievable American system that we have that allowed you to thrive. Somebody invested in roads and bridges. If you’ve got a business — you didn’t build that. Somebody else made that happen. The Internet didn’t get invented on its own. Government research created the Internet so that all the companies could make money off the Internet. The point is, is that when we succeed, we succeed because of our individual initiative, but also because we do things together."

This is undeniably true.  And the particular phrase that the GOP is harping on (out of context) — "you didn't build that" — referred to "roads and bridges".

It should be noted that even Romney agrees with that.  Romney recently said:

I know that you recognize a lot of people help you in a business. Perhaps the banks, the investors. There’s no question your mom and dad, your school teachers, the people that provide roads, the fire, the police. A lot of people help. But let me ask you this, did you build your business? If you did, raise your hand. Take that, Mr. President.

Here’s how Obama described the relationship between “the people that provide roads, the fire, the police” and business owners in that speech Romney cited:

In short, Romney: Yes, “a lot of people” help business owners, among them government, but in the end, “you build your business.” Obama: “We succeed because of our individual initiative, but also because we do things together.”

How is that any different?  It's not!

Now the Romney campaign has come out with a blatently dishonest ad on the Obama quote:

… except that it is heavily editted to take the Obama quote out of context.

This is how Obama’s speech is presented in Romney’s version:

If you’ve been successful, you didn’t get there on your own. You didn’t get there on your own. I’m always struck by people who think, well, it must be ‘cause I was just so smart. There are a lot of smart people out there. It must be because I worked harder than everybody else. Let me tell you something. If you’ve got a business, you didn’t build that. Somebody else made that happen.

Here, once again, is Obama’s own speech. Note the sentences immediately preceding and following “you didn’t build that,” in which the president explicitly tells the audience he’s talking about roads, bridges, the Internet and other public infrastructure investments that are all but universally considered a boon to the private sector. The sentences cut out of Romney’s ad are in bold. 

If you were successful, somebody along the line gave you some help. There was a great teacher somewhere in your life. Somebody helped to create this unbelievable American system that we have that allowed you to thrive. Somebody invested in roads and bridges. If you’ve got a business — you didn’t build that. Somebody else made that happen. The Internet didn’t get invented on its own. Government research created the Internet so that all the companies could make money off the Internet. The point is, is that when we succeed, we succeed because of our individual initiative, but also because we do things together.

Mitt, if you have to engage in heavy editting to make Obama say what you want him to say, that alone means that he didn't say it in the first place.

P.S. Astute political observers will note that Obama's statement was really something akin to something Elizabeth Warren said last year:

"You built a factory out there? Good for you. But I want to be clear: you moved your goods to market on the roads the rest of us paid for; you hired workers the rest of us paid to educate; you were safe in your factory because of police forces and fire forces that the rest of us paid for. You didn't have to worry that marauding bands would come and seize everything at your factory, and hire someone to protect against this, because of the work the rest of us did…. Now look, you built a factory and it turned into something terrific, or a great idea? God bless. Keep a big hunk of it. But part of the underlying social contract is you take a hunk of that and pay forward for the next kid who comes along."