When DS was 11 months old, he spent 24 days in two different hospitals, had an ambulance ride to larger children's hospital in the city, four doctors, tests, tests, tests... Final diagnosis: severe pnuemonia and an unidentifiable (I'm not joking, they were mystified) blood infection of a type normally seen in critically ill patients with no immune system?????

The larger hospital charged $800 a night for the bed alone in a four-bed unit.
We maxed out our insurance completely ($5,000 plus antibiotics which our insurance did not cover). I didn't complain, the total bill would have been over $50,000. The only bright light was that it was February, so we didn't have to pay for any other medical expenses that year, like tubes he got for ear infections, and the 8 months of follow-up x-rays until his lungs were fully healed.
It took a year to pay off the bill and we had to cut EVERYTHING at our house. No satellite, cell phones, internet, eating out. One car.
We learned our lesson, within another year of hard work, we paid off our car (three years early) and got ourselves out of credit card debt. Now we keep an emergency fund for this type of thing and have better insurance.
Talk to the hospital about a payment plan, this worked fine for us. We got a monthly bill and no nagging or phone calls. Worked out fine. I'm not sure I would use a credit card to pay off a medical bill, not with the interest rates being jacked up. Also, you could look into the tax deduction for medical expenses, but you may not qualify. We tried and were about $1,000 shy of getting it.