Why can’t single-payer and private insurance co-exist?

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yes, under the British system you can get supplementary private insurance (normally if you want to cut down on waiting times for non-urgent procedures - the NHS has an 18 week waiting list policy, but recently that policy has been abolished for things like cataract removal and hip replacement surgery); it also allows you a bit more choice and private rooms. Even those with private insurance will still go for the NHS in serious emergency situations and chronic diseases like cancer though.