Let me try to rephrase me previous post to make the point more clearly, and -taking off my William Jennings Bryan hat- with less populist righteousness. Cost-of-living measures attempt to look at quality adjusted price changes. When the rents in a given city increase, the housing portion of the CPI goes up and by that measure so has the cost of living. But this increase can simply reflect the positive amenity value of the city going up; less crime, better parks, etc. If you really wanted to do quality-controlled measure of cost-of-living you would control for the fact that higher priced cities reflect higher amenity values.
So when people say well $250k isn’t a lot when you look at the cost of living in that area, that higher cost of living does not reflect complete controls for quality but in fact measures the higher amenity value of the city. This is just consuming a better, and so more expensive good.