My Dad never wore his ring when I was growing up, and he still doesn't now.
I remember noticing this one day and asking my Mum about it. She said he just never wore it because he worked as an Electrician and it wasn't safe. That makes sense. It never bothered my Mum that he didn't wear it.
Now, years on I am married and my husband doesn't wear his ring. He wore it for a few months after we got married, but he used to take it off when he went to work (similar situation to my Dad- it wasn't safe to wear it in his job), but then he'd put it on at home or when we went out. Problem was it annoyed him. It used to slide down his finger and he'd be playing with it all the time. In the end, he stopped wearing it cos (a) he was scared it would fall off and (b) it didn't fit properly and so it annoyed him. Resizing wasn't an option. We had it sized when we bought it, but I guess he just has a funny shaped finger or something, because once it was over his knuckle it was too small. The next size down wouldn't even go over his knuckle. It never bothered me, but he used to keep it in his top drawer beside the bed- and that showed me it was special to him.
Then one day our house got brken into and his ring was stolen from that drawer. We have never replaced it. So, although I always wear my ring, he doesn't wear his anymore.