It depends, Random. There are certain times when they aren't allowed to do that. If you lose your job and are paying out on COBRA (which means continuing that insurance by paying directly to that plan, and is usually pricey) and sign up for an individual plan, they don't have you fill out the health forms (at least, they didn't have me do it - my husband had to fill one out). It's complicated, I'll grant you, and not everyone has a lot of options.
One hideous problem that I can tell you we have here in my area is with "charity care." This is when some low-income person with inadequate insurance (or none) runs up a hospital bill. The hospital, any hospital that receives any kind of public funding, has a separate fund for helping these patients; they often require the patients to sign up for social programs if they qualify, but then the hospital forgives large chunks of the bill and works out a payment plan with the patient.
However, some hospitals are
hiding this from patients, or at least not going out of their way to let patients know. If the patient doesn't know their rights, the hospital collections people may come after them with a vengeance and try to go after any property they might have. One local hospital used to have a small office dedicated to providing this service with a few posters up to help low-income patients know that they should ask about this option, but the office was closed and the posters taken down.
Private hospitals tend to make pretty good money, but they often hide it in ongoing construction and landscaping projects. Overpaid hospital administrators get sent to Tahiti for pricey health care conferences and things like that all the time. Meanwhile, poor patients get squeezed.
Despite the scare stories, I see that my local hospital has a
web page all about charity care and payment options. It says they paid out over $10M last year on it.
Overall, most Western countries seem to have some form of a two-tier system, or are moving in that direction. Those who can afford it go private, those who can't get assistance. Canada was supposed to be a true single-payer program but more Canadians are pushing for more private options (and they want the options
in Canada, which makes every kind of sense).
There are a lot of problems in American medicine, sure. If you ask me, the place to
start solving them is actually with medical school. Medical school here is brutal and expensive. I'm not saying it should be easy and cheap, but to me it seems like an area that could use some changes. As it is, who would go through all of that if they couldn't make a seriously good living when they are done? If they don't they can't even pay their six-digit student loans.