But a three-judge panel of the 5th U.S. Circuit Court of Appeals ruled last year that Burdine's rights were not violated, despite the lawyer's inattention.
Now the full 15-member New Orleans-based court is revisiting the case, hearing arguments Monday.
"I keep praying and hoping justice will be done," Burdine said last week from Texas death row, where he's been imprisoned 16 years for fatally stabbing his lover with a butcher knife. "I can't say which way it will go."
Burdine, now 47, came within moments of execution in 1987 before receiving a court-ordered reprieve. He denies killing 50-year-old W.T. Wise at a trailer they shared in the Houston area in 1983.
At a 1995 evidentiary hearing, three jurors and a court reporter testified that Burdine's now-deceased attorney, Joe Cannon, often napped for as long as 10 minutes at the trial.
Burdine's current lawyers called the dozing attorney "no more sentient than a potted plant."
A federal judge ruled Burdine did not receive a fair trial and ordered the state to retry him or set him free.
But the 5th Circuit ordered the former nurse to remain in prison while it considered the case. In a 2-1 opinion, the panel said the conviction should stand because Burdine was unable to show that Cannon slept during critical parts of the trial.
Burdine's attorneys appealed to the full court. "Supreme Court precedent guarantees counsel at all critical stages of a criminal proceeding, including, of course, the trial itself - the most critical stage," lawyer Robert McGlasson argued in a brief preceding Monday's hearing.
Burdine's case has bolstered arguments that capital punishment is unfair - and legal representation inadequate - for defendants who can't afford to hire their own lawyers.
"This isn't the only time a defendant's lawyer has slept through portions of a death penalty case in Texas," says CBSNews.com Legal Consultant Andrew Cohen.
He points out that "it's obviously a problem and it is also a symptom of a problem, particularly in states like Texas where they simply don't pay court-appointed attorneys enough to always ensure that criminal defendants are adequately represented under the constitution."
"You would think that common sense and decency alone would require there to be some clear rule that says that if your lawyer falls asleep in the middle of your capital murder trial, you automatically ought to receive another lawyer, or another trial, but that isn't yet the case," adds Cohen.
Cohen explains that in order for Burdine to actually get a new trial, his new legal team will have to prove that "the lawyer's conduct - the fact that he was asleep - actually made a difference in the outcome of the case and that sometimes...is hard to do."
Last week, a measure was filed in th Legislature that would halt executions in Texas until 2003 while a commission studies issues such as legal representation. Texas has executed two prisoners this year after a record 40 executions last year.
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