The BA isn't "full" by any reasonable measure. See this list of US cities by population density for comparison: https://en.wikipedia.org/wiki/List_of_cities_by_population_d.... San Francisco is 21st on the list, at 17,246 people per sq mi; the next densest city in the Bay Area is Daly City, which is 49th at 13,703 people per sq mi.

By contrast, NYC has 27,016 people per sq mi; the densest suburb of Boston (Somerville) has 18,431; West Hollywood has 18,297.

There's plenty of room for more people in the Bay Area, if denser housing is allowed.