So What's the Problem?
Wrong. It happens all the time. And we mean all the time. In one famous 1989 case of a rape of a jogger in Central Park, five men went to jail after the cops got them to confess to the assault. Then they were released, after it turned out they didn't do it (another guy already serving time for a different assault finally came forward).
"Only the guilty have anything to fear. And the innocent, if we're having a slow day."
It turns out it's actually not all that hard to get a confession out of an innocent person. The same high-pressure psychological techniques meant to wear down a guilty suspect will make a lot of innocent people confess to something they didn't do. Since innocent people are more likely than criminals to waive their right to remain silent, they're put in a high-stress situation where they're not even clear about what they're being charged with. They also may feel guilt for some unrelated reason (they saw the crime and failed to report or stop it, for example). So, they say whatever they need to say to make the interrogation end.