After having my blood drawn little water bumps appeared down my arms is that normal???

Not normal.: If these bumps are itchy and did not appear until 1-3 days after the event, the most likely problem is from a contact allergy to the adhesive bandaid applied on the needle site. If the problem spreads or persists > a week, see the doctor. You mentioned "down my arm"- if the blisters spread way beyond the bandaid site, then the problem may be unrelated to the blood drawing.