It is not uncommon to commute 30 miles or more to work (one way) in Los Angeles, for example. My commute used to be 62 miles each way, then I cut it to 45 miles, and now it's about 10 miles. There is no effective public transportation system that could take me to/from work faster than even driving in rush hour traffic could.
So at least in LA, it is definitely a big problem to not have independent cars.
Edit: I should also say that where I used to live in Europe, those commuting distances would take me to completely different cities!
LA is all about the neighborhood you live in. LA neighborhoods are basically self-contained cities. if you choose the right one, you don't need a car. e.g.: santa monica, venice, west hollywood, downtown, silverlake, dt culver city, dt pasadena, etc, etc, etc. these are dense urban areas with top notch city housing within walking distance (10 minutes) of everything. it's city-expensive but not like NY or SF (with exceptions, like santa monica).
most people in LA commute clear across the entire county, which is why they need a car. most of these people are middle class, they could move if they wanted to, but they don't want to.
today, with uber and other services, the only reason you would "need" a car in LA is to commute, unless you make terrible living choices. also the expo line (downtown-west side light rail) will be done in a year.
30 miles one way doesn't sound uncommon in Tokyo as well, though the extensive public transportation system makes the need of cars much less than US. So I think it also depends on the policy of city design. I guess it's harder to omit cars for the rural parts of US rather than cities.
(I lived in LA about a year without car. It's ok if you're single, but if you have family it's a totally different story.)
So at least in LA, it is definitely a big problem to not have independent cars.
Edit: I should also say that where I used to live in Europe, those commuting distances would take me to completely different cities!