FAA spokesman Ian Gregor reports an American Eagle Embraer E 135 came within three miles of the tail of a Chilean-based LAN Airlines plane on January 19.
Pilots are required to be five miles apart so smaller planes can avoid wake turbulence that can send them out of control.
Gregor called the error "serious" but said there was no imminent danger on the smaller plane. A spokesman for the controllers union says the controllers are overworked and understaffed. Gregor says the error was not caused by understaffing.
The Associated Press contributed to this story.