The Vault is split into two sides.
One side is pretty much a normal vault, tons of normal people, food generation a la the Sierra Madre's vending machines, pure water supply yadda yadda yadda.
The other side is like the other one, full of nourishment and entertainment.
Except it is home to one man.
His side of the vault is completely sealed off from the other vault, except for a secret system that will unlock a door to the other side of the vault.
This code is obtainable by thoroughly studying the 'Vault Resident's Manual' which, if checked for patterns in the text, tells him to pull out the books of a bookshelf in a certain order, before hearing the message the overseer gives him. He then has to figure out a second code and tell it to the overseer, and once he does, the vault sides are joined.
If the man dies before he can figure it out, a boy is taken from the populated side by the overseer when it is 4 years old, and given to the empty side and looked over the overseer's Mrs Handy, a special Mr Handy designed for mothering children.
Test: To see if a man can be so driven to escape from loneliness that he will search desperately for companionship.
Another one would be that a man is put in a vault designed for one person. It has everything he could possibly want, but only a single bedroom, toilet etc.
However, the overseer is an AI that has a quirky, happy, interesting personality, and is designed to keep him company.
Through the use of unique drugs the man has to take every year, the man can live indefinitely.
(Don't ask me how it's possible, just roll with it)
The test is to see if the man can stay completely sane with a relative life of luxury and a friend who he can only talk to.