(The following answer is in the context of Fallout 3)
Vault 101 (where you start) is obviously functioning as intended at the beginning of the game, and continues to function more or less as it should throughout the course of gameplay, depending on the player's actions. The big exception, of course, is that people can leave and re-enter Vault 101 under certain circumstances.
All of the other vaults the player encounters may or may not have living residents in them, but all have experienced events that are departures from the public understanding of what their function was supposed to be. That is, to keep citizens safe and unharmed for a predetermined period of time until the earth was safe to live on again (astute players will gather intel through gameplay that hints at a more sinister purpose). In that sense, none of the vaults besides Vault 101 are functioning as intended.
You may want to examine the list of known vaults and read their descriptions for further information.