Can you, in general, keep exactly the same insurance contract if you change employers (but paying it by yourself)? Because if not, and you ever had any serious

Can you, in general, keep exactly the same insurance contract if you change employers (but paying it by yourself)? Because if not, and you ever had any serious condition, then, I would consider it tied to your employer. In other countries, pension plans can be "tied" to your employer, i.e., while employed by X, you get some extra perks in your pension plan (and the payments into the pension plan are deducted from your salary, so you are paying for it yourself), but if you move to another company (or become unemployed) you cannot keep the same pension plan (even if you want to keep paying it by yourself); you are forced to accept whatever conditions are available at that time.