The story in New Vegas wasn't deep. It was cliche on top of cliche, likely on purpose to give it a pulp adventure feel. That's totally fine but it's strange to hear it praised as anything special. The left for dead cliche (a western staple), rich old man reaching for immortality cliche, and then the lame "resolved via hidden super-weapon" cliche. A typical video game story intended to move the player from point to point.
In fact, it's really hard to think of many video games at all that can be called "deep". One that does come to mind is Suikoden 3. Fantastic concept but probably too ambitious for the console technology at the time. More than that, though, I think it stumbled on the way to the ending, even if the ending itself was fantastic. It was like it was missing segments that should have allowed the player to connect to the conflict. I don't know if it was rushed but it didn't set up the ending as well as it could have.