Hard boiled eggs can be eaten after 10 days as long as the hard boiled eggs have remained refrigerated for those 10 days.
Also make sure the hard boiled egg still smells good and if the hard boiled egg smells good then it's safe to eat and should not harm you.
Or if the hard boiled egg smells bad or has a bad taste to it then throw it out as it could make you sick.
I have eaten hard boiled eggs that I've had in the refrigerator for 10 days and never got sick from it.
But I made sure to smell it first and see if it were still good.
If you leave the hard boiled egg out at room temperature too long then they can go bad quicker but the refrigerator keeps some bacteria from growing so fast on the eggs.