I have anemia and my nail beds are purple a third of the way up. Is this because I'm anemic?

Yes. Yes but did they tell you what is causing your anemia?
Anemia. Weakness, tiredness, sometimes shortness of breath, pale skin, pale membranes in the mouth, and nail bed duskiness - these are common changes with anemia, and become worse as the severity of the anemia gets worse. Make a plan with your physician on how you are going to correct the anemia.