"Jett was the most wonderful son that two parents could ever ask for and lit up the lives of everyone he encountered," said the statement posted on John Travolta's Web site. "We are heartbroken that our time with him was so brief. We will cherish the time we had with him for the rest of our lives."
A house caretaker found the teenager unconscious in the bathroom Friday morning, police said.
"A nanny attempted to revive him. All attempts were made, but he couldn't be revived," Travolta's attorney Michael Ossi, who was also in the Bahamas, told ABCNews.com Friday. "They tried as hard as they could to revive Jett."
An ambulance took him to a Freeport hospital, where he was pronounced dead.
The family, including Jett's 8-year-old sister, Ella, had been celebrating the New Year in the Bahamas.
Ossi said the teenager "has had seizures in the past, but they were controlled. This one couldn't be."
Jett's health made national news in 2002 when his mother disclosed that at age 2 he had a poorly understood condition called Kawasaki syndrome, a collection of symptoms that stem from swollen arteries.
Kawasaki syndrome primarily affects children younger than 5, though it can occur in older children. According to the Centers for Disease Control and Prevention, about nine out of 100,000 children have Kawasaki syndrome. The incidence is higher among Japanese and Korean children, though the syndrome can occur within any ethnicity.
Kawasaki syndrome expert Dr. Robert Frenck, a professor of pediatrics in the division of infectious diseases at the Cincinnati Children's Hospital, said the syndrome, however, is not usually associated with deadly seizures -- especially in children who have already recovered from Kawasaki's, which he said is a temporary condition.
"If there's a major complication, and if someone dies from it, it is a [coronary] aneurysm," he said. "It doesn't happen frequently, but that is what we really worry about. ... That can set the kids up for a heart attack."
The New York Post and other media outlets have in the past published unconfirmed reports that Travolta's son had autism, though the family has always maintained that he suffered from Kawasaki's. Autism is associated with seizures.
"There is a relationship between autism and seizures; as many as 40 percent of children and young adults with autism may experience seizure and adolescence is a particular time of vulnerability," said Dr. Bryan King, the director of child and adolescent psychiatry at Seattle Children's Hospital's Research Center for Health Services and Behavioral Research.
"There are hormonal changes that could increase the risk of seizure, and certainly there are ongoing brain changes that take place during adolescence, but no one knows why the risk increases in older children."
What little information is available on autopsy results further suggests that Jett Travolta may have been an epilepsy sufferer. If this is the case, he could have died from a massive seizure that led to a condition known as sudden unexplained death in epilepsy patients, or SUDEP.
Dr. Orrin Devinsky, director of the New York University Comprehensive Epilepsy Center, said SUDEP "is a relatively common problem among patients with uncontrolled tonic-clonic -- a.k.a. grand mal or convulsive -- seizures. In patients who have these frequently over a 10-year period, the incidence of SUDEP may be 8 percent or higher."