If you just say "ある", it usually means "sometimes". If you add "よく", "しょっちゅう", or something like that, you can say its frequency in more detail.

I would say " (ずっと寝ていないことが) よく (ある)" or " (ずっと寝ていないことが) しょっちゅう (ある)", which means "I am OFTEN in a state...", "I am CONSTANTLY in a state..." or something like that.

If you use another particle "が" instead of the original one "の" and say "ずっと寝ていないこと [が] ある｡", it makes sense. ／Even in that case, it means "I am SOMETIMES in a state of staying up all night.", which might be a bit different from what your English sentence means.