WASHINGTON — The Supreme Court struggled Wednesday with a life-or-death question for Hank Skinner, a Texas death-row inmate: Whether a defendant convicted long ago has a right to new DNA testing of old evidence.

Skinner, who was an hour away from execution in March when the high court intervened in his case, was convicted of a triple murder that happened on New Year’s Eve in 1993. Long after his conviction he asked to have DNA tests on much of the crime scene evidence, including two bloody knives, a towel and the swabs and hair samples taken from one of the victims, Twila Busby.

District Attorney Lynn Switzer refused to allow the testing, saying the request came too late. Skinner’s defense lawyer did not ask for the DNA testing at the time of his trial, once it had been shown that his client was covered with victims’ blood when he was arrested.

Skinner appealed in the Texas state courts and federal courts, but he could not win a ruling ordering the further testing. Texas, like the other states, adopted a new law in the past decade that permits some DNA testing, but a judge ruled Skinner did not qualify because his trial lawyer opted against the extra testing.

“”We are seeking access to evidence that has never been tested,”” said Robert Owen, a Texas law professor, representing Skinner.

Students from Northwestern University’s Medill School of Journalism investigated Skinner’s case and found evidence that pointed to Busby’s uncle as the possible killer. She had left the New Year’s Eve party after he had made “”crude sexual advances”” to her. Skinner was seen dead drunk on a couch next to where she was killed.

Last year, Skinner sued in a last minute bid to have the DNA tested, but he lost in the lower courts.

During Wednesday’s arguments, the justices sounded split and uncertain of how to proceed. Several of them noted the court has frowned upon opening the door to whole new appeals at the end of a long case.

“”How do you get around Osborne?”” asked Justice Sonia Sotomayor. Just last year, the court in a 5-4 decision ruled against William Osborne, an Alaska rapist, who unsuccessfully sought DNA evidence to rebut his earlier conviction.