In the European Union, there are strict regulations considering health insurance:
You have to be health insured (remember that in unemployment stage, most governments pay your basic health insurance).
If you are employed in any country, you have to be insured in such country. If you are employed in more, you can choose any of them.
However, the regulations don't say what to do if the new country is unable to register you instantly. In France the registration takes up to several months. Most people choose to pay the minimal insurance in their original country until the French administration proceeds their file. However, this is officially illegal.
As far as I know, the countries don't imply any fines if you break the rules, but the health insurance companies can probably void the insurance if they find out you break the rules.
Example 1: You are a 27-year-old PhD student in Czechia and you come to study in France for an exchange program for which you obtain a "bourse du gouvernement français" which is not a work contract. You can choose any of these two countries (and theoretically any other country as well) for your health insurance. In the Czech Republic, as a student 26+ you pay circa EUR 40 a month for that.
Example 2: You graduate your masters in the Czech Republic and you get a PhD contract in France. As long as you don't work in the Czech Republic, you have to re-register in France, since work contract is a work contract, even if you're also a student.
(Example 1 is actually my situation, Example 2 is the situation of my good friend.)