Mary Agnes Moroney (1930)
Catherine Moroney was a struggling mother when a social worker responded to her request for help. The worker, Julia Otis, offered to take her daughter two year-old daughter Mary for two weeks to help her financially. Moroney agreed; Otis wrote once and never returned. The whole con was made possible by an oversight: The newspaper that printed Moroney's request for help also included her address, which the paper normally didn't publish.
Gedhun Choekyi Nyima (1995)
This (then) six year-old was recognized by the Dalai Lama to be the eleventh incarnation of the Panchen Lama. Shortly afterward he was whisked away by the Chinese government, who named a different boy as the official incarnation. Since that time Nyima is reportedly under house arrest. All requests to see him have been denied by the government, who offer no proof of his whereabouts.
Boston Corbett (1888)
Corbett, the soldier who shot and ultimately killed assassing John Wilkes Booth, was later sent to a mental institution as he suffered from increasing mental illness. He broke out of the institution in '88, though where he went to and how much longer he survived are unknown.
Charles Rogers (1965)
Rogers remains the only suspect in the murder of his own parents, having fled the scene before the grisly crime scene was discovered. He has since been declared dead, though no trace of the intelligent recluse was found. What makes his case really bizarre are the alleged ties he has to those believed to be involved in the JFK assassination, though those connections are debated.
Frederick Valentich (1978)
The twenty year-old Valentich was flying over the Bass Straight when he reported being stalked by a strange aircraft. Part of his plane may have been found (being narrowed down to a group of planes which would have included his Cessna) but after his radio communication ceased he himself (and his craft) vanished. Some have interpreted the craft he described to be a UFO (Valentrich was a UFO enthusiast himself) and assume he was attacked or abducted. Another group claims it was a staged disappearance. The most likely explanation is that he got disoriented and crashed.