Single biggest clue as to the start date. Todd Howard during the E3 presentation: "Now we're not going to be spoiling or talking about the game's story tonight. But you do survive and get lowered into the vault. Events transpire, but you then emerge 200 years later as the Sole Survivor of Vault 111."
The first part is really important. Anything story related that is influenced by the date, such as Project Purity expanding and finding its way further North, the involvement of CWBoS, including an older and more capable Arthur Maxson, as well as key NPCs such as Madison Li establishing themselves within the Commonwealth, all such elements will also be required to be kept under wraps. And the best way to stop people from accurately guessing is to throw them a red herring with the "200 years" comment. But there is a further clue that ties the story and the start date together, and that is the huge number of references to the American Revolutionary War. It is my belief that the events seen during Fallout 3 pertaining to Harkness (the Replicated Man quest), including the mention of freedom and independence, signify the start of a new Revolutionary War. A war between humans and synthetics. And this would have begun as soon as the story in Fallout 3 ended. This would also have required a certain amount of time to pass, in order to for such a conflict to progress to the point where it the dominating event in the Commonwealth. This would not have influenced the storyline of New Vegas, because of the distance involved, but it would most certainly have become common knowledge in the Capital Wasteland, and so a number of other familiar NPCs and factions will almost certainly show up. The most logical time for us to emerge from the vault would be during the later stages of this conflict, so that we can assist not only with fighting, but also with ending the war. And given the length of the first Revolutionary War, the most obvious time for this to take place would be around the middle of 2285, giving the entire conflict a duration of 8 years.
But looking further into Todd's quote, we can see even more evidence. The final sentence suggests the point of view of the protagonist. He or she believes that approximately 200 years have passed due to the dates on the various logs, as well as the date shown on the Pip-Boy itself. It would be perfectly natural to round the actual figure in your head so that it's easier to digest, rather that repeatedly telling yourself, "I can't believe I've been under for 207 years, 8 months, and 11 days." They also consider themselves as being the Sole Survivor of Vault 111, due to them being the only person left inside the vault who is not dead at the time. We know from the Brian T Delaney interviews that behind every quest and every in-game action, they are searching for their family. This means that they were not in the vault at the time the protagonist awoke, and so their whereabouts are unknown. And if one half of this sentences can be regarded as being ambiguous, then why not the other?
Pete Hines will also say whatever Todd wants him to say about the date, because it is his job to support his team with either promoting or protecting any specific details about the game. And as for Codsworth, he is an NPC coded by Bethesda Game Studios, and so he will no doubt say whatever he is told to say. As an in-game character however, his precise comment about the temperature of the protagonist's coffee was made under relatively 'normal' circumstances before the Great War. Given his humanised personality, the two centuries or so following the nuclear holocaust would have almost certainly had some effect, especially if much of this time was spent on his own. When he finally sees his former master, he is not passing him a beverage, he is reuniting with somebody from his distant past. This is why his reaction could be construed as being highly 'emotional'. And yet, if we look at the dialogue options after he offers to 'whip up a snack' (probably the first meaningful task he has been able to carry out in a very long time), there is one that says "200 years?" I am more than certain that, if we choose this option, Codsworth will say something along the lines of "Well, it's actually 207 years, 8 months, and 11 days to be precise, but after such a long time one can easily lose count."