It is pretty clear now that Half-Life 3 will most likely never happen but most fans believe that Valve did actually pick up the game and have started developing it in the past.
However, according to GameInformer, an anonymous Valve source reveal that Half-Life 3 has never formally existed which means there was never a Half-Life 3 in Valve. However, the source did reveal that there were actually people coming out with stories and material for the game when Half Life Episode 2 was completed.
But when it came to the time to get serious, the sourced said that Valve had a bunch of big projects all lineup and Half Life 3 just got swallowed up. He also added that the third episode was just wasn’t working with Gabe and at the end, those that wanted to bring Half-Life 3 back found that they time would be better spend if they work on other projects.
That was how Half-Life 3 slowly ceased to exist anymore which is such a pity since the game had such a huge following. You can read the full interview here.