Earlier quoted context omitted.
Boosting housing by building vertically is virtually unlimited. The trickier part is supplying the added people with water, electricity, gas, food, and other supplies without augmenting infrastructure (which is failing left and right as it is) and convincing them to never travel out of or into the city using its already-at-capacity freeways and rail.
For the last part, having mixed-zone development is essential. Put housing above supermarkets! Offices inside apartment complexes! Make it easy for people to walk around. I think that people don't need convincing not to use their cars... it's more that the space needs to be built in a way to where cars are not necessary.
Sure, that's the ideal and gradually becoming more common. (Heck, the Two Worlds development in Mountain View -- ground-level retail with residences above -- dates to the 1980s.)
But again, the trick is figuring out a way to force people to only take jobs within their building and/or move to the building where their job is (and move both when either changes). It just doesn't work that way in the Real World.