actually they are denied all the time. Just because one is sick does not mean they are entitled to MRI's, or X-Ray's, or cancer treatments. The ER's only can do so much. Plus many without health insurance don't see doctors and in many cases it's to late - a recent study by Harvard stated that close to 43K die a year because they have either no insurance or are under-insured.
Many people who show up at free clinics have insurance with but with high deductibles