Both Kyle and Jessi undergo vigorous and extreme training. Tom Foss is Kyle's trainer, based on his military background. Each exercise is a study in focus as well as physical effort. In this story, it is not physical effort that produces a nose-bleed, it is mental exertion directed to something outside themselves, of the sort that only Kyle and Jessi can do. We can assume that such effort raises blood pressure in the head to the point of causing particularly Jessi to have a slight nose-bleed.
Jessi does her early training by herself, jumping down from her window ledge, running, climbing back up the building. When Brian Taylor starts training her, we see her first nose-bleed when he tells her to move an empty glass while running on a treadmill. He has attached sensors all over her, but both of them ignore the warnings.
After sinking over three-hundred balls at pool, Jessi won't stop unless Kyle beats her. He is about to sink the 8-ball to win, when Jessi makes the ball deflect enough to miss. Her nose is bleeding, but a fight starts when Kyle tries again to get Jessi to leave. Outside, she shows Kyle a bad cut on her palm. As she makes it heal, as she imagines Sarah could do, her nose bleeds again. Kyle warns her that pushing herself too hard too fast is what killed Sarah. After saying "I'm fine," Jessi collapses from total exhaustion.
When Kyle hugs Andy the day before she goes in for tests about her cancer treatment, the lights flicker. After she leaves, he sees in the hall mirror that his nose is bleeding, like Jessi when she tries too hard. What he did for Andy, he did not mean to do, and it takes a few days for him to recover.
Kyle learns from Declan where Jessi is training in the woods. When he finds her, she is straining to levitate two rocks from a tree stump, one more than Declan saw her doing. Kyle's arrival breaks her concentration, and she wipes off the blood as she turns to him.
The next occurrence is when Brian wants her to lift a small dumbbell which she says is too heavy, but he insists.
Adam warns Kyle that Jessi will destroy herself with over-exertion. What sort of damage is done that the bleeding is only a symptom of, we do not know. Whether it is simply the misery and self-pity Jessi feels when her mother does not come to meet her, or whether she wills her own system to kill itself, the strain again shows before she does collapse in death at JP's Diner. By the time she falls onto the floor, blood has dripped down from both sides.
Jessi is not in a situation of such continued and excessive effort during the rest of the story.