Yeah sometimes the brake surface of the rim might rub onto the brakes itself because the brakes might be too close to one side of the brake surface.

You might have to center the whole brake caliper itself if that's the case. Does it only rub when you tilt the bike in one direction?

If you hear/feel some rub in both directions, try to loosen the tension of the calipers again, check if they're centered, and then try it out again.