It's the question we get more than almost any other, usually asked a little sheepishly. The honest answer: yes, but probably not for the reason you think.
Brushing cleans the surfaces of your teeth, but it misses the tight spaces in between — which is exactly where decay and gum problems tend to start. Flossing isn't about being thorough for its own sake; it's about reaching the part of each tooth your brush can't.
If flossing makes your gums bleed, that's usually a sign to floss more, not less — though if it keeps up, it's worth a check-up. Written the way we'd explain it in the chair, and structured so it turns up when someone searches it at 11pm.