Hairdresser Refuses to Bleach

Even if your hair looks healthy, it might have hidden damage or underlying issues that make bleaching risky. If your hair has been colored or chemically treated before, is naturally very dark and you're trying to go very light, or has a fine, delicate texture, then it’s more vulnerable to breakage. These are all factors that a stylist takes into account, especially if their job is to help you keep your hair, not just change its color.
As for how likely it is that something will go wrong: without seeing or feeling your hair, it’s impossible to give you a number. But if a professional who’s actually examined your hair is saying it’s a high-risk move, that’s worth taking seriously. She’s likely trying to prevent major damage, not holding you back because of a lack of skill.
That said, if you’re unsure, it’s perfectly reasonable to get a second opinion from another experienced stylist. You might also want to talk through alternatives like highlights, balayage, or a more gradual lightening approach. And if you’re set on going blonde eventually, ask about ways to strengthen and prepare your hair for it over time.
Remember: a good stylist doesn’t just say "yes" to everything. Sometimes the best professionals are the ones who tell you "no" - because they care about your hair's long-term health. For your information, below is some additional explanation about the possible harmful effects of bleach on hair.

See also: Where does the color go when you bleach dark hair?