As everyone by now should be aware, Canada has universal health care. How this happened is not as well known and can provide an example of how apparently piecemeal systems can end up providing universal coverage.
Saskatchewan began the process in 1946, but couldn't provide universal coverage for all its residents because it simply didn't have the money to provide for free hospital care for everyone. In 1950, Alberta followed with a similar program, but they too didn't cover all (they did reach 90%) of their population.
The carrot came in 1957. The federal government offered to pay 50% of the costs of programs which covered hospital and diagnostic costs for the provinces that created them, so long as the programs met five conditions: public administration, comprehensiveness, universality, portability, and accessibility. By 1961 the lure of federal money resulted in all 10 provinces agreed to participate and set up programs, if they hadn't already.
Saskatchewan led the way again and used the money that had been freed up by the federal contribution to provide complete, universal coverage for basic health care. The programs in place covered the hospital and diagnostic costs of the majority of the population, but there was one major gap: the actual cost of seeing a doctor was not covered, and the Saskatchewan government had decided to close it. The Saskatchewan Doctor's Strike resulted in 1962, as doctors feared that single-payer would make them government employees and affect what they could earn. While the strike was deemed a failure in that Medicare was not dismantled, doctors did get higher fees, assurances of professional independence, and more seats on the Medical Care Insurance Commission which would determine things such as fees and what would be covered. The result was that by 1965 a solid majority of doctors favoured the medicare system.
The success of single payer universality in Saskatchewan was such that in 1966 the Federal government introduced the Medicare Care Act, which extended the existing program so that it would pay 50% of the costs to any province that created a universal, single payer health care system. And thus we arrived where we are.
The important thing about this is that it all didn't come at once. The federal government used the carrot approach to get provinces to participate, and provinces always had the opportunity to opt-out if they had so desired (by not opting in). No one forced any of the provinces to participate, but in the end the political pressure to do so from their own voters became unbearable for any government who didn't want to.
And so it could be down there.