A 3.x to 3.9 series should be a pretty safe change. The issues may be more related to the things you are not be telling us.
--Where is your theme located? If you took a standard theme, say twenty thirteen or twenty fourteen and brute force updated it without child theme, your site is guaranteed to break upon update. The update will overwrite those files and you will lose all of your customization. If you haven't done so, take 12 minutes to understand how easy it is to use a child theme.
--Wordpress core is pretty robust (relatively backwards compatible) for recent changes. What may not be so friendly are the plugins (and theme) you've got. Who knows how they will react?
--While not exactly related to a wordpress update, what is minimum PHP version from your server host? Hostgator just made an update that affected many of our customers. (I found errors in plugins using firebug/firefox & chrome developer tools that weren't visible to normal customers. We filed bug reports with the plugin authors involved. All fixed now.)
--Absolutely best thing is to make copy of your site, templates and plugins. Test on local host first, then go ahead and update on production site.
--Plugins can be a concern. You might try turning them off first on the local site, update wordpress, then turn them on one at a time. You will probably have to check each plugins author to see if there are updates available. I'm finding out that many plugin authors do NOT use the update notification system.
--Frankly the biggest risks I've seen from one click updates are a zealous admin can simply click once and update a site without first testing. Way too easy to do. We've added the words "REMOVE_ME" in front of critical update API keys in the admin panel so we don't get a one click (oops) update.
--Of late my only update failures have come from my purchased template suppliers. I've got one source who has updated their template three or four times in the past six months withOUT backwards compatibility. At each update the site crashes, and my customer is upset. I've finally had to stop all updates for that template (which could be a risky thing if ever a security flaw is found... been there before...) I would abandon that supplier and ask for my money back if I didn't already have lots of time invested in that theme. No way to undo it now. Bad choice on going with those guys.