When a government’s legitimacy starts to break down, the government might be in trouble because

no legitimacy = no credibility. If the government is legally elected or nominated, and there is no way how to remove, all the system can be blocked. If there is a constitutional way how to remove it, there is no problem.

Explanation: there are many political systems in the world. Always depends  where is the legitimacy of the government derived from. From the legislativy assembly (Parliament, Congress) or executive (president) ? In presidential systems its legimitimacy is derived from  the president (the USA, Latin America), in parliamentary systems it usually requires approval of the legislature.