In general, yes. However it's also dependent on how good your workplace is. I've worked in places that are good (luckily most my jobs) and some that are terrible. In bad places a small to medium fuck up will straight up get you nixed. At good places they'll be understanding, especially if you go to them with idea on how to fix it as well. I work in tech & I've seen people get fired for server outages after unexpected traffic spikes (this was before server scaling was relatively easy). At one of the good places I worked I accidently deleted a user table on a production database on the first day of my boss's holiday, losing about 20K paying subscribers. I immediately called my boss and we talked about the best ways to fix it for now. Upon his return he took me out for a coffee & I was sure I was a goner but my boss told me it was an expensive lesson for both me (being more careful) & them (having faulty backup & rollout procedures). Later that day our tech team had a meeting to look at what processes & checks needed to be implemented to stop it happening again & in the unfortunate event it still did happen, how we could reduce it's impact. This was in a relatively small startup in the mid 2000's, proceeded to work for them for a few more years.

I freelance how but with my main client if something goes wrong the first thing I'll do is call them & we'll spitball the best solutions until we come up with a good fix. Once fixed we come up with a prevention plan.