They said a far more common problem is hospital employees leaking information, like the Los Angeles hospital clerk who pleaded guilty to snooping into the files of Britney Spears and Farrah Fawcett and selling the information to a tabloid.
"It's my strong impression that hospitals generally carry out their responsibilities," said Jim Walden, a former federal prosecutor. "It's often the opposite where contrary to privacy rules there's a leak of one kind or another."
Burress sustained a gunshot wound in the thigh when his .40-caliber Glock semiautomatic pistol accidentally discharged at a Manhattan nightclub early Saturday.
He was taken to NewYork-Presbyterian Hospital/Weill Cornell Medical Center by a teammate. While Burress has been suspended by the New York Giants and faces criminal prosecution on gun possession charges, the hospital's role has come under scrutiny as well.
Burress checked in under an alias - Harris Smith - and was released after nearly 11 hours. The hospital never notified police, which they must in any case involving a gunshot wound.
Mayor Michael Bloomberg said the district attorney should "go after" the hospital's management. For his part, Manhattan District Attorney Robert Morgenthau said he wants to "get to the bottom of why this wasn't reported."
A hospital worker was suspended for not reporting Burress' injury. She apparently arrived at the hospital at 2 a.m. to treat him, but it's not clear why she was called. The Giants say she has no connection to the team.
Ken Sunshine, a publicist whose clients include Barbra Streisand and Leonardo DiCaprio, said it's common for celebrities to use false names at hospitals, as they do at hotels.
"People often use assumed names to maintain some privacy," Sunshine said. "There are a lot of very crazy people out there, and it's no secret that celebrities are targets."
The challenges facing celebrities seeking medical privacy were highlighted this week when a former employee of the University of California at Los Angeles Medical Center pleaded guilty to selling information from the records of patients including Spears, Fawcett and California first lady Maria Shriver to the National Enquirer.
Defendant Lawanda Jackson worked as an administrative specialist at the UCLA hospitals and began using her supervisor's password to access medical records inappropriately, authorities said.
Prosecutors said the Enquirer deposited checks totaling at least $4,600 into the checking account of Jackson's husband.
The case is just one example of celebrities' medical privacy being violated. Stars such as Elizabeth Taylor, Christina Applegate, George Clooney and Patrick Swayze have all had medical records leaked to the media.
Actress Jennifer Garner told USA Today, "I've been at the doctor's office and had another person in the waiting room call and report that I was there - I think it's appalling."
As for the reverse - a hospital failing to meet a legal obligation to report a gunshot wound, as apparently happened with Burress - that is rare.
Ron Kuby, a lawyer who has represented Gambino crime family associates among others, said gunshot victims may go to a trusted family doctor who will keep it quiet, but a major New York City hospital can be expected to follow the law and report a shooting.
Even if the patient is rich or prominent?
"Generally speaking," Kuby said, "rich and prominent people don't end up shooting themselves in the leg."