During the episode "The Impossible Astronaut," we see the Doctor get shot by River Song. He starts to regenerate, but then he gets shot again, killing him. But including the war Doctor, the Doctor had used up all of his regenerations, so how is it possible for him to regenerate? We know he was inside the Teselecta and that he didn't actually die, but if he did die he wouldn't have regenerated. My question is, why bother to fake a failed regeneration that couldn't even have happened? Why not just fall to the ground?
In-Universe: The Tesselecta didn't know about the regeneration limit and The Doctor never bothered to tell them, or anyone for that matter, until he revealed it to Clara in Time of the Doctor. As a result, when they were "shot", they acted as the real Doctor would have (given the information they had on file), mimicking the process of regeneration before it was interrupted by a second shot. His companions, especially the River watching, would have been suspicious if he hadn't started to regenerate.
Out-of-Universe: Moffat, as usual, didn't think that far ahead. Part of it was that "The War Doctor" wasn't even a thing in 2011 (when those episodes would have been written), since the BBC were still hoping that Eccleston would come back for the 50th anniversary special. After it became apparent that he wasn't going to forgive them that easily, 'The War Doctor' came into being.
Another part of it was that nobody even bothered to think about it until Matt Smith announced his intention to leave Doctor Who, at which point fans started making the point of "Oh, but he's hit his limit, because War Doctor and Tenth's 'extra' regeneration".
It could easily have been written that there is no limit, or that the limit is bigger than 13, or maybe even that Tennant's "handy" extra regeneration didn't count, but instead Moffat decided to write every piece of fan speculation on the internet into the show as hard fact.
Personally I believe 11 really did die the original time but he fixed it. I also believe that what seemed like a regeneration starting would have fizzled out as a dud regeneration even if he had not been shot a second time (or was secretly the tessalecta). Of course opinion that it was a dud regeneration is based only on later learning he was on his last incarnation secretly. It may be notable that Capaldi's doctor stated that a timelord's dead cells will continue to try to regenerate.
I believe upon discovering his death - 11 created an elaborate sequence of events that ultimately kept the timeline intact by creating a perfect replica of his original death when the universe was looking. These sorts of shenanigans are probably why the Silence were going nuts because he was clearly alive and dead at two points in time and the universe was allowing both timelines to exist. Note the second time River refused to shoot and broke the universe and then the third time I say it was the Tesselecta. Of course I just put way more thought into this than Moffat did.