Health Care Bleg

Since my sister has so kindly forced many of you to read It's Better Left Said (or at least link to it), I thought someone might be able to help me with a long standing question I have. The argument I see most commonly made against universal health care is that it results in "long lines and waiting lists." Perhaps. But it always seemed to me that the long lines were caused by the addition of millions of potential patients to the system. So what "long lines and waiting lists" really means is that some people who can afford private health insurance who used to get treatment quickly would now be replaced by some people who could not afford private health insurance. Am I wrong? Is there another cause I am missing? I am no expert on health care and insurance, and am curious.