I'll give you my most recent example on a shirt I have from Mexx. The reason for bleaching it was because I got a spot on it while picking up something else I was bleaching - forgot to change out of my good shirt into my work stuff. Doh!
Anyway, it was a grey shirt with white and burgundy stripes on the shoulders (a different patch of fabric, one on each shoulder). 100% cotton if I recall. The bleach turned the grey to pink. Turned the burgundy into red and brightened the whites.
Next I went with the Rit color remover. It turned the now pink body of the shirt to a creamy white, turned the red to yellow and the white remained white. The shirt actually looks quite excellent now and it's like I have a brand new shirt - it's already a few years old, so the refreshed colors are a nice change.