Yes they can. Just a couple of days ago I saw a case where a woman got the exact same coloring she always has used for years at her stylist. She had a horrific allergic reaction that can occur at anytime in any of our lives, even with products we've used previously with no effect.
Her face is severely swollen, her eyes are barely slits in her face, she has scabs and patches all over her head and her hair is falling out. Her doctor said that a commonly used chemical is the cause of all this and reiterated the fact that allergy test should be done before every service to make sure that the person is not allergic. Beware!