For a long time, Xbox 360 was more popular than PlayStation 3 and seemed to have a good chance of outselling Sony. But eventually, Sony picked up steam, and by the end of the generation, they had sold more than Xbox 360 worldwide (thanks in large part to much greater success in Japan, while Xbox 360 was slightly bigger in the West).
It's difficult to say exactly when this turnaround began, but perhaps it was when Microsoft started focusing on Kinect and casual games, and chose to almost exclusively release new instalments in its core series: Fable, Forza, Gears of War, and Halo, something that led to a franchise fatigue. Another educated guess would be when Sony released Naughty Dog's sequel to Uncharted: Drake's Fortune, namely Uncharted 2: Among Thieves. This cemented Naughty Dog as a giant and proved what the difficult-to-develop PlayStation 3 was capable of in the right hands.
Now, the ever-well-informed YouTube channel DidYouKnowGaming reveals that Microsoft definitely took note of the Uncharted series and actually wanted to challenge it with its own alternative, which went by the working title Project Ranger. It was developed by the wizards of graphics Black Tusk Studios, which later changed its name and became the Gears of War studio The Coalition.
Microsoft believed strongly in the game, which was briefly shown off at E3 2013 (though without a title or formal announcement), and the premise is described as a Mission Impossible-inspired adventure.
We don't know exactly why the game was shelved, but The Coalition went on to develop Gears of War 4 instead, and Xbox One never really managed to compete with PlayStation 4 in terms of sales.
Check out the video below. It's a very interesting feature with several glimpses of Project Ranger that give an idea of what it would have been like.
<social>https://www.youtube.com/watch?v=8mT_AwsKOtw</social>