DS only liked one type of 'binky'. I tried to take it away from him at 12 months, but he started in on his thumb...back went the binky! At 18 months he started at a new daycare. Since he only liked one kind of binky, I gave daycare 2 and had 3 at home. Well, he lost both of his daycare ones, and the daycare provider bought some replacements without mentioning to me that he needed more. Well DS didn't like the new ones, and was napping fine without them. So, after our dog at the last one at home that was it, I think DS was about 19-20 months old.
He did try sucking on other baby's binky a couple of times, but he did fine at night with out it.
DD on the other hand, would not use a binky. In fact she wouldn't take a bottle until I went back to work when she was almost 4 months old.

She sucks her thumb, it started in the hospital and I couldn't stop it. It was either me (breastfeeding) or her thumb. Since I didn't relish the idea of being the human pacifier, I gave up trying to stop the thumb sucking. Now, at almost 3, she's sporting some buck teeth and callouses on both thumbs. While I knew DS wouldn't go to kindergarten with his binky, I can't cut off DD's thumbs.......

Oh well, like someone else posted, braces are cheaper than therapy.