I live in Canada, and here's how my last couple months with the healthcare system has gone (anecdote and first world problem alert):
About two months ago, it was the beginning of a weekend and there were no clinics open that would see me until the coming Monday. Something was rapidly deteriorating the skin on my hands. So I called an advice line, and the nurse on the line said that I should go to an emergency department.
So I went to an emergency department, waited a few hours and was then seen by a doctor who could not diagnose me. She gave me a prescription for a week course of antibiotics and a topical cream, and said that a dermatology office would call me over the next week. I filled the prescription on my (thankfully exceptionally good) employer drug coverage. I followed the prescription regimen for the week and did not hear from the dermatology office, thankfully the cream was enough to apply according to the directions for up to a month and was seemingly keeping the deterioration of my hands at bay. I waited another two weeks for the dermatologist to call, and they did not. I called the hospital and was connected to a representative which did not speak or understand English very well and ended up being connected to the wrong department; then I called the hospital again and the lady at the hospital connected me with the dermatology office which was to receive my referral. After about 25 minutes I was able to deduce that the ER had not yet faxed the referral to the dermatology office, so I called the ER and thankfully still had my discharge form on hand to reference and help them track down the referral. I made sure they faxed the referral, and confirmed with the dermatology office which told me that I would be scheduled for about Tuesday last week (and would receive a call when it was confirmed). I still haven't heard from the dermatologists and I'll probably have to nag them again. I'm currently attempting to treat the condition by myself by reading random articles and blogs, and medical research on Baidu Wenku (thank god for high IQ) and doing okay, though it's not eliminated.
For critical/emergency care, Canada's system often works quite well (though sometimes fails horribly). If you have anything less than an emergency, or need dental work or drugs, you need your own insurance. That said, the fields which are still private in Canada (lasik, dental, optical) are reasonable to pay for by yourself if you're middle class. The fields which are covered by government insurance are illegal to pay for by yourself.