Doctors for John W. Hinckley Jr. said yesterday they believed he will make a full recovery from what apparently was his third attempt to kill himself since his arrest nearly two years ago on charges of shooting President Reagan and three others.

Hinckley, 27, who was found not guilty by reason of insanity last year of the attempted assassination of the president, was found semiconscious on the floor of his room at St. Elizabeths Hospital Sunday morning, apparently after taking an overdose of his own medication, according to knowledgeable sources.

The incident is being investigated both by St. Elizabeths and by Washington police. A spokesman for St. Elizabeths said yesterday that officials are trying to determine if Hinckley may have saved some of his medication and then taken an overdose, but "the investigation is still going on and we don't know exactly what happened."

Hinckley was rushed by ambulance to Greater Southeast Community Hospital where he was first listed in serious condition and placed on a respirator to assist his breathing.

Dr. James Levy, president of Greater Southeast, said that by late Monday afternoon the toxic level in Hinckley's body was only 20 percent of what it had been when he was brought in Sunday.

A hospital spokeswoman yesterday said he was listed in fair condition, that he was "alert and oriented," and that he had been taken off the respirator and was breathing on his own. "We do believe that he is out of the woods." she said.

Hinckley is allowed to drink liquids but is also being fed intravenously, the spokeswoman, Debbie Carson-Gorman said. "He is able to talk, but according to the people caring for him he is not saying much of anything; he is being very quiet."

Doctors at Greater Southeast would not say what substances Hinckley took, although laboratory reports have been completed. Carson-Gorman said that "the patient and his lawyers will not give their permission for the hospital to release that information."

Knowledgeable sources said they believe Hinckley may have taken an overdose of an anti-depressant drug that had been prescribed for him and that he had been saving up the drug.

One source familiar with Hinckley's situation said Hinckley apparently was "up and about" on Sunday morning and took the drugs that morning. "It was not a situation where he was left all night without aid," that source said, adding that it appeared Hinckley suffered no permanent injuries as a result of the incident.

In May 1981, two months after he was arrested for the shooting, Hinckley took an overdose of the aspirin substitute Tylenol and the tranquilizer Valium while he was being held in a federal prison at Butner, N.C. Six months later, Hinckley attempted to hang himself in his jail cell at the Army stockade at Fort Meade, Md.

According to testimony at his trial, Hinckley had attempted suicide twice before his arrest.

At the trial, Dr. William T. Carpenter, director of the Maryland Psychiatric Research Center, a defense witness, said Hinckley's mind "was filled with thoughts of homicide, suicide, death and the end of the world" in the months before the shooting of Reagan.