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Technically yes, the terms under which a Schengen visa is issued require you to have health insurance up to a certain amount of coverage. You may be asked for proof of this in the form of health insurance documents when you're entering the Schengen zone and/or at any passport control checks.
In practice though, it is unlikely that you will be asked to provide documentation for this. However, travel health insurance is still a good thing to have as hospitalisation costs in case something does happen can be expensive. You don't need to get an expensive policy; at least in the UK, you can get short term travel health policies which cost less than £10. Even though these policies may have high deductibles, for serious accidents they will still help you recover medical expenses.
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I have never been asked for that on European Airports Check points. They usually ask you
I once saw a police asking a person to show him how much money this person got. It really depends on how the police look at you…
However, travel insurance is a requisite to enter a Schengen country. In fact, it is stated in every European embassy webpage. It costs about $50 USD for two weeks covering $50,000 USD. I believe you should spare no expense. No one ever knows.
Almost at the bottom of the Wikipedia article on the Schengen area, there is a table of procedures border guards follow when checking travellers who cross external borders of the Schengen area.