Yes sesame seeds do eventually go bad.
Properly store sesame seeds will last for up to 4 years though before they do actually go bad.
To make sesame seeds last a long time you should store them properly which helps to also better retain flavor and potency, store in containers with tight-fitting lids.
After 4 years the sesame seeds may be okay to eat but they will not be as nutritious or taste as good.
Sesame seeds are best tasting and most healthy when they are at their freshest.
So it's best to eat the sesame seeds within a year of purchasing them.
However you can keep sesame seeds longer than a few years and they will be okay.