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Basic and supplemental health insurance do not necessarily have to be bought from the same provider. Health insurance providers may not terminate supplemental insurance policies if holders decide to switch providers for basic health insurance. They may, however, charge an additional fee to cover higher administrative costs that arise when basic and supplemental insurance are taken out with different providers. It is also possible for your provider of supplemental insurance to cancel any discounts you have had so far (family discount or collective discount). Only few insurers actually do this, though.
Here's what to do if you wish to buy basic and supplemental health insurance from different providers: