The following is based on Van Buren and has not been confirmed by canon sources.
This vault's experiment was devised by scientific genius Derek Greenway. Most of the parents were either "accidentally" redirected to other vaults or were in the early stages of health conditions that would no doubt cause them to die soon after entering the vault. Instead of a human controller, Vault 29 would have a ZAX super-computer. The ZAX would be programmed to raise children with the aid of robotic helpers, educate them in a primitive culture, and upon their reaching maturity, release them into a controlled environment. They would then be free to rebuild society from the ground up.
At one point Greenway explained his plans for Vault 29 to Diana, the human brain connected to a powerful computer, to see what her opinion would be. He was quite surprised when Diana was appalled at the idea. She said that, although the idea had merit, she found it to be morally objectionable on several levels. She recommended that the plan be scrapped. When Derek refused to do so, Diana took it upon herself to become involved with the project without his knowledge.
When the Great War started and civilization began to fall apart, Diana appropriated control of a satellite dish and aimed it at Vault 29. She then transmitted a series of security codes to the vault's ZAX unit, and gained control. Over the years that followed, Diana was able to see to the upbringing of the children, who founded the village of Twin Mothers outside of the vault. Every year she ordered a worker robot to leave the vault under the pretext of checking conditions outside. In reality they were preparing an area to receive the vault populace when they were ready to leave. Lastly she had a series of projectors installed at strategic locations, ready to project her chosen image. Finally, when the vault doors were opened the inhabitants found Twin Mothers built. Everything was coming together, Diana was ready to play god.
By 2253, the vault itself has ceased working, but it's still a holy place to the inhabitants of Twin Mothers.
Twin Mothers and Vault 29 did not appear in any of the published games, but were to appear in Van Buren, the canceled Fallout 3 project by Black Isle Studios. It is not certain if it will appear in any future Fallout game. Vault 29 was mentioned in Fallout and the Fallout Bible.
Early Van Buren leaks suggested that the Twin Mothers Vault would be 31 rather than 29. It is unknown whether the number was changed, or a mistake. 
The Twin Mothers Design Document and the Fallout Bible contradict each other as to whether Adults were accepted into the vault. The Bible indicating it is children only with adults diverted elsewhere whereas the design document suggested that older parents would be accepted with infant children.