DOCTOR: She sustained a minor hairline fracture to her wrist.
LORELAI: So she broke her wrist? ….
DOCTOR: It’s a tiny fracture, absolutely nothing serious. I’m gonna put a cast on it. She’ll wear it for a couple weeks, that’s it.
There is essentially no difference between a fracture and a break – a hairline crack and having a bone shattered into pieces are both referred to as a fracture. The terms are interchangeable. Fractures usually take 6 to 8 weeks to heal, a hairline fracture may be on the shorter side of that, but only two weeks in a cast doesn’t seem plausible. However, it’s not long until the end of the season, which probably has a lot to do with the doctor’s treatment plan!
Lorelai insists that the doctor do some more X-rays, which he agrees to, but hairlines fractures don’t typically show up on X-rays, so it’s a waste of everybody’s time. It’s just to give Lorelai a chance to keep Rory busy while she goes off to do some yelling. Most parents wouldn’t leave their injured kid at a hospital in the middle of the night like that, but this is TV, not reality! Unfortunately for the plot, it makes Lorelai look incredibly selfish. I mean, more so than usual.
The Annotated Gilmore Girls 