Hard boiled eggs will keep safely in the refrigerator for up to a week before going bad.
As long as you keep the hard boiled eggs refrigerated during that week the eggs will still be safe to eat and if you want to you can mark the container you put the eggs in with the date you boiled the eggs to tell whether they are still safe and good to eat.
You could pickle the hard boiled eggs and keep them longer if you can do so you'd have some good pickled eggs which are delicious and make a great healthy snack.
I've kept pickled eggs for months before without them going bad as I snack on them throughout the month.