Did Lee Harvey Oswald go to jail? Oswald was arrested, but not for the assassination of the  President. On November 22, 1963, about 45 minutes after the  shooting at Dealey Plaza, Oswald had been stopped by â¦Dallas police  officer J.D. Tippit for questioning. He shot and killed the officer  at point blank range.    Shortly thereafter, Oswald was arrested as the suspect in the  shooting of the police officer. He was in possession of a .38  caliber revolver (which ballistics later matched to the bullet that  killed Officer Tippit).