The Enclave, the shadow government of the United States - a conglomerate of business and political leaders who saw the nuclear fires raging on the horizon and sought an escape.
As for the Overseers - most of the Overseers likely figured the experiments would never touch them in the case of the more malicious experiments. But a lot of Vaults collapsed due to outside influence the Overseers couldn't contain - such as Vault 15's overpopulation, or the Necropolis' Vault door being designed to not seal properly. Vault 22's end seems to be a genuine accident. Vault 87....FEV was still a unknown quantity when the Vaults were designed, so obviously countermeasures failed in that regard.
Then there are Vaults that managed to beat the odds - such as Vault 21, where every decision was made by a game of chance and they actually were a stable society until House came along and swept them away in his move to secure the Strip.
Then there were the Vaults that were unintentionally successful. Vault 13, for example, was never designed to maliciously fail (Such as Necropolis' Vault or the twisted fantasy of Vault 112), but the water chip malfunction was designed to see how they handled the problem. Vault 15 is another case of this - while the Jackals, Vipers, and Great Khans all spun out from this seething mass of people, Vault 15 also provided the core settlers of places like Shady Sands and, as a result, transformed into one of - if not the - most successful power to rise out of the ashes of the Great War.