All vaults were filled when Nukes started hitting. You can hear the few holotapes (Keller family or something), about a family trying to seek shelter in 101, 87 and others, but failing to do so.
"We couldn't get into a Vault. 101, 87, didn't matter -- all full to capacity."
- Keller family transcript 2.
This confuses me somewhat, though it was probably an oversight on Bethesda's part. Many of the vaults weren't full to capacity because of 'cry wolf' mentality of the constant siren drills prior the actual bombings, so when the sirens did sound for the official alert, many people stayed put, there's not a chance they would have been able to reach a Vault once the bombs dropped, Fallout 3's world map is (understandably) misleading, it would take days to travel between locations. Vault 13 was probably the only vault that was exempt from this cry wolf mentality, because it was the very last to be constructed, even later than it was planned to be finished, and so the drills for that vault were considerably less before the bombings.
Update/Sources: Fallout bible 32069 March.
Vault 13 is finally completed - it is the last of the Vaults, and drills begin. Due to its late completion, the "cry wolf" effect that hurt the other Vaults is not as pronounced.
2077, October 23.
Great War - Bombs are launched; who struck first is unknown... and it is not even known if the bombs came from China or America. Air raid sirens sound, but very few people go into vaults, thinking it is a false alarm. The Vaults are sealed.
EDIT: Vault Wiki states 101's mission as "Evaluation of performance of an omnipotent Overseer in a closed community "
I find it odd since it's impossible to predict that a omnipotent Overseer will succeed the first one.
Well I get the general idea that the vault experiments were destined to fail, I think the real experiment was in the severity of the results, rather than the actual experiment itself. There is a mention of this in the Fallout Bible, albeit a very small one, that one vault's experiment, similar to another, was destined to fail before the other, which implies they were both expected to fail, you could then extend that logic to many if not all the other experimental vaults. There were only 17 vaults that were made to work the way they were designed, and these I think, were the only vaults expected to succeed in any positive way.
Update/Sources: Fallout Bible 1# Vault 55 - All entertainment tapes were removed.
# Vault 56 - All entertainment tapes were removed except those of one particularly bad comic actor. Sociologists predicted failure before Vault 55.