It should not be possible, but these things happen. In 1986, a Texas jury convicted Michael Morton of killing his wife. The local sheriff and a prosecutor read a note he'd left to his wife on their bathroom mirror about their sexual problems and concluded he must be the perpetrator when Christine Morton was found murdered in their bed room. The note read:
At the same time, detectives ignored or withheld numerous bits of evidence that might have suggested Morton's innocence. Someone tried to use his dead wife's stolen credit card. The couple's four year old son described "a monster" coming into the house and killing his mommy. Police never bothered to collect a bloody bandana left behind the house.
Fortunately for Morton, he was sentenced to 25 years to life, not death. It took 25 years for lawyers to force the state to test the bandana for DNA, a test that turned up another suspect. Subsequently the District Attorney's files revealed the other bits of evidence exonerating Morton that had never been given to the defense. The man whose DNA showed up on the bandana had killed another woman nearby a year after Morton's wife. In their haste to convict the wrong man, authorities had let a real monster kill again.
This entire story was sensitively narrated by NPR's Scott Simon this Saturday. This 14 minute clip is absolute worth hearing in its entirety.
Yes, there are good reasons why it seems dangerous to allow the state to execute convicted criminals. What if the verdict is wrong? It can happen.