Cases where their clients confess are probably the most difficult for defense attorneys. When a defendant confesses, juries almost always convict. The psychology at work here is obvious: Why would someone admit to something capable of landing them in prison unless they really did it? The proposition that only the guilty (and perhaps also occasionally the insane) confess is a common sense conclusion that seems all but inescapable.
But despite appearances many innocent (and generally sane) people have falsely confessed, and far more often than hardly anyone would imagine. A recent study found that of three and eleven people conclusively proved to be innocent by DNA evidence, over 25% had given false confessions.
For a fascinating discussion of police interrogation techniques and why they have a often produce false confession, consult this article from the New Yorker: http://www.newyorker.com/magazine/2013/12/09/the-interview-7