Yeah, this sounds like a stiff link(s) problem to me. You don't need to replace the chain but you do need to loosen those links up to be flexible. Stiff links can be a serious problem if the chain doesn't run through the rear derailleur smoothly. If a stiff link gets stuck in the derailleur it can be fatal to the derailleur and even the rear triangle of the bike's frame. I've seen it happen.
Al