I just thought that I'd add my thoughts to the current frenzy of analysis and speculation regarding the Fallout 4 Teaser Trailer. Take this as seriously or as lightly as you wish:
The leaked script, if accepted as accurate, confirms that the protagonist is either the father or the mother in the family we see in the trailer (I suspect that if you select a female character, then the protagonist is the mother with a suitably-modified script and a female VA). My guess is that he/she is an employee of Vault-Tec (thus the Vault-Tec poster in the kitchen/dining nook) and thus had a guaranteed place in Vault-111.
More importantly, the protagonist is a veteran of the war in Alaska and the annexation of Canada (a soldier for a male character, an army nurse for the female character, given that the 1950s-themed pre-war society would likely have still had rigid gender role bars). Something of a 'dark past/the redeemer' aspect is suggested to me by this possibility.
Given the leaked script suggests that the inhabitants of Vault-111 were kept in cold sleep until the subjective 'present', I'm thinking that they were part of Vault-Tec's sick social experiments. Specifically, they were the control group - kept in Cold Sleep to ensure no social changes occurred in the group so that the monitoring staff at Vault-112 could compare how an unchanged social group managed post-war compared to the various changed groups in the other vaults.
I suspect that the timer that was meant to wake the inhabitants of Vault-111 failed and they were not awoken until a second-level fail-safe trigger event - the activation of a Garden of Eden Creation Kit nearby.
Consequently, instead of being catapulted into the post-war anarchy or even the rebuilding epoch of Fallout 3 and Fallout - New Vegas, the inhabitants of Vault-111 are in a world where society is mostly-rebuilt. There are large population centres, fairly organised government (the Enclave on the East Coast and the New California Republic on the West Coast and into the interior) and probably a lot less official and local tolerance for strangers rocking the boat on the grounds that "we didn't used to do it this way".
BTW - The people on top of the vault when the blast-wave hit all survived. Look at the art of the Vault's top-cap and you'll see lots of high-voltage electrical insulator discs set around the circumference. I suspect that the top cap had an electromagnetic field designed to protect those on top of the cap from the blast waves of reasonably distant nukes long enough for them to get inside. That doesn't changer the horror that those fourteen survivors will always feel as they remember seeing the desperate people just outside the field getting incinerated and blasted apart as the wave hits.
As the game is set in Boston, near the Institute, I wonder if they're going to be the bad guys or at least the plot drivers of the story. Most of it will be about the Old Soldier learning to find his or her place and one for the survivors in the Vault in the new world that has grown up in the new Commonwealth.
However, I suspect that The Institute may know more about Vault-Tec's experiments than has been previously admitted. Even now, centuries later, those brilliant minds, freed of law, ethics and possibly even flesh and blood, might be carrying on those social experiments to create a perfect society. When the new society does not meet their predetermined standards, they may decide to destroy it. Fortunately, the Old Soldier is there to stop them.
I strongly suspect that The Enclave will be strongly entrenched in Boston (Commonwealth City?). The presence of Eye-bots suggests that they keep important locations like Diamond City under observation and the sequence when you're flying in a Vertibird with a Steel Legionnaire suggests that they also provide at least a significant semblance of police services within the civilised parts of Boston. Outside those, as the sequence with the feral Ghouls, the Super-Mutants, the Mirelurk and Deathclaw, the Massachusetts Wastelands remain a dangerous place.
That said, apart from the nuke that hit nearby New York, the area seems to have escaped the worst devastation. I wonder if the main attacks on the area were chemical rather than nuclear, promoting mutations rather than outright destruction of the sort we saw in earlier games.
This story will be very strongly plot-driven with certain tasks you need to complete in order to progress further. We've seen that there is going to be a home-base (the garage set, which I suspect is in the protagonist's home district). I suspect that you'll be able to craft, repair and upgrade. Amongst the things you can do is craft and customise weapons, armour and other survival equipment, improve your PipBoy and fix the family's Mr Handy service robot. The long-term-project is to get a set of T45 power armour, a minigun and a rocket launcher and heavy lasers operational so that you can storm your way into the Big Bad's base.
Collectables are back in the form of training and recreational magazines that may also give skill bonuses as well as the Bobble-heads with a similar function. There are also the collectable cards on the Vault-Tec noticeboard with the perk symbols on them, making me wonder if you can optimise skill and perks that your character has over time. Possibly, skill levels can be greatly modified by removing old cards from the collection board and adding new ones (although doing so may erase the old bonus until you earn the perk again).
I suspect that the opening scenes of the Trailer is a slightly-redressed version of an in-game scene where the protagonists visits his old home and has a flashback to The Last Day. However, it is also possible that, as with The Last of Us, there is going to be a prologue/tutorial first act of the game set at The Beginning of the End of the World.
Diamond City is going to be a shopping district, IMO. You can buy weapons, food, survival equipment and medicines there.
The rest of Boston is going to be strongly story-led. There will be places you can go to regularly like the bar ('Memory Pit', IIRC) and the bank built under USS Constitution. The latter, I suspect, will change almost any unit of exchange, especially precious metals and other rare materials, for bottlecaps. However, some you will be only able to access in the story mode when they are part of the story. These may include the Enclave's local HQ in one of the new skyscrapers we see rising over the old city.
Beyond that, I suspect it is going to be pretty much a standard post-apocalyptic exploration/survival game with a very, very significant story aspect. There will be a multiplicity of weapons, including melee weapons, craftable or available from Swatters at Diamond City. There may also be a significant stealth element and puzzle solving and 'find your path' elements that opens up different concluding options towards the end of the story. This latter option interests me as it means you may have to play the game many times and make multiple different decisions to open up different and even optimal endings.
I also wonder if there will be a 'sandbox' game that unlocks when you complete the story mode?