While bleaching and dyeing hair may cause your hair to break off at the root and fall out, this happens only in some cases. For the most part, the occasional hair treatment will not cause permanent damage.
In some cases, bleaching or dyeing your hair can cause damage to the hair. This usually occurs if the treatments are performed too closely together. Bleaching then dyeing your hair can strip it of moisture and damage the follicle, cause the hair to either fall out from the root or to break out somewhere along the shaft.
A lot of the time, when hair treatments go badly, it's because they were performed improperly. If the hair is dyed too frequently or with too high a concentration of dye, you may notice a loss of hair immediately after the treatment. If this sort of hair treatment is continued on a repeated basis, it might result in continued hair loss, which may be noticeable as thinning or even bald patches.
Don't Rule Out Other Conditions
Even though you bleach or dye your hair, that might not be the cause of your hair loss. Other things like genetic hair loss, wearing the hair in constricting styles or even taking certain medications like birth control pills can cause hair loss. If you're concerned about your hair falling out, discontinue any harsh chemicals you're using on your hair and consult with a dermatologist.