Indeed they can!
Having considerably low ferritin can cause fatigue and depression, among other symptoms. If they stay too low for too long, it can lead to iron-deficiency anemia.
HOWEVER, many people find that having "reduced" or even mildly "insufficient" iron stores (that is, ferritin levels) does not affect the way they feel. "How low is too low" depends on the individual. Personally, I found that I started feeling really bad when my ferritin levels went down to 12 (according to a standard blood test).
If you decide to start taking iron supplements, please be sure to get your ferritin re-tested every few months. You wouldn't want to accidentally cause your iron levels to go too high, which can also cause problems.