I'm pretty sure that a lot of people thought about it when they played the vanilla version, but it's even more obvious now with the Far Harbor DLC, where one of the main NPCs, Dima, asks what the Sole Survivor's oldest memory is.

Surprisingly, you get to choose between the bombs dropping and spending time with your family (pretty much a few minuted before the bombs dropped). It's astounding that those are the choices you get and that Dima also suggests that not remembering your childhood friend and stuff from before that is a clear sign that you're a synth with implanted memory. But let's think of some of the facts that we've learned so far, that make this claim sound like it has a lot of sense.

The beginning of the game

So the game starts with a rather dream-like scene where the Vault-Tec approaches your home and you talk about the bomb shelter you gained access to in case of the bombs falling, and suddenly... they do. Kinda strangely condensed, even for something that's supposed to make you jump into the game as soon as possible, right? But if you think about it, the Vault Tech and the Robot butler are key parts for places you'll later visit on your way to the Institute (weird, huh?).

Getting out

You're out of the vault as the sole survivor, but did you stop to think what happened to the others? All of the vault dwellers were put into hibernation, but it was only a family of three that survived being frozen? Or maybe none of the vault dwellers survived, and the Sole Survivor was placed there as a synth, with implanted memories, making it look like they survived?

The robot at the house

You get out of the vault and find the 200 year old ruins of your home with... your butler robot inside, working fine and without any visible damage. Could it be he was put there by the Institute to guide you? 200 years and no raiders or scavengers took fancy of a robot butler? I mean, I bet a thing like that would be worth a lot of caps.

The dog

It's pretty obvious that Dogmeat is a synth dog. It  can survive bullet wounds and explosions. Another thing that makes it pretty much obvious that it was the Institute that puts him in your path is that after you leave Kellogg's place and you're about to start searching for his trace, Dogmeat suddenly is waiting at the door as you get outside. The dog is also able to trace Kellogg's scent right into his hideout miles away. His scent, riiiiight.

The Vault Tec

It's another one of the suspicious characters that somehow got to survive the bomb drop to meet the Sole Survivor after 200 years. Let's think about it. Guy survives a blast like that and gets turned into a ghoul to live for 200 years, with no friends, so that he could meet the person he talked for a few minutes and suddenly become long time friends. What is he doing in Goodneighbor? He's been put there to scout for the Sole Survivor, of course. It's the place to go after you want Kellogg's memories scanned by Doctor Amari. The vault dweller goes to have a rest at the hotel and the synth ghoul is waiting there for them, reminding that they talked for a few moments 200 years ago and that he's the same person you remember. Or better yet, they've implanted into your memory.

Paladin Danse

At some point you learn that Danse is a synth and the Brotherhood wants him hunted down. The question is - when was the original Danse replaced by a synth? I'd assume that it happened after you both go out on a mission to that place where you cleared the building from old generation synths. After he gives you the laser rifle as a reward you part ways. He is likely grabbed by the synths and taken to the Institute, to be then replaced.

Sean is not Sean

If you paid attention to what Sean was saying, you probably noticed that his version of the events is a bit fishy. If he's your son the Institute abducted and raised as part of their system, who then became the Father and leader, then why is there no kids or pretty much families in the Institute itself? It looks like a recently made place designed for the takeover of the Commonwealth. There's people who are experts in certain fields, that would suggest that they were sent there to do a certain job as well. Sean is likely an actor or a synth. His role is to put the synth vault dweller in charge. Most of the council members are well aware of that, some aren't. Sean is suddenly dying, even though the Institute is capable of putting his brain inside a synth, or likely replace his organs. Weird, isn't it? Another thing is you teleport inside the institute and walk around in a power armor and with your guns, yet noone seems to care. There were no safety measures placed on the way. Pretty much anybody would be able to capture a courser and build the device to teleport their assassin there. Yet they knew exactly what was going on.

So what happened? My guess is that the Institute needed an ambassador. They implanted memories to the Sole Survivor synth that made them think their baby was abducted. The memories involved one of the families that didn't survive being frozen in the Vault. As Sean explains, it was his will to let his parent out to "see if they survive." It's a ridiculous reply that would only suggest that the person who was hired to pose as Sean was really bad at improvising a quick reply. The plan worked well - the Sole Survivor has to infiltrate the RAILROAD and also becomes friends with people in Diamond City. Then a chance to infiltrate The Brotherhood Of Steel emerges and is botched terribly when Danse's switch becomes apparent. The main character is a synth placed by the Institute to take hold of the Commonwealth. An ambassador that makes the impression of a fellow human willing to help the people, not knowing that they are just a puppet for the Institute (that can unknowingly rebel against them).