What does someone mean when they say that they have "incredibly weak nerves" ? It does not refer to a medical condition.
It really, really depends on the context.
This means "Please don't tell me anything that might prove stressful for me":
PAMELA[whining]. Did you have to tell me they're all out there? I've got weak nerves. [faints]
This means "Don't be a smartass; I'm pretty irritable today":
PAMELA [threateningly]. Look, buster. You know I've got weak nerves. I can snap any moment. You keep annoying me like that, I'll strangle you.
The following means "She'll be getting on my nerves all the time. She's got a high-pitched gravelly voice. I hate her":
PAMELA[philosophically]. If Linda's going, I'm staying right here. I realize she's a riot and everything, but I've got weak nerves. I'm entitled to get some rest: I'm on vacation.
Because I am a musician, the first thing that comes to mind is stage fright. However, this generalizes to any case where a person is easily upset, frightened, or made nervous.
So a musician could say this:
I have very weak nerves!
This means that they easily get nervous on stage, like in a performance or for an audition.