I have really cut down on saying "I'm sorry" at work. I got to where I felt as if all I did was say, "I'm sorry, I'm sorry," because lots of things I have to do to make people better are at best unpleasant and at worst actually painful. I also used to apologize profusely for wait times and for physician delays and so on. Recently I've decided to stop that. For one thing, it really seems to increase patient anxiety if they think there are that many things that I NEED to be sorry about. For another: the ER is not a fast-food establishment, meaning we try our level best to treat patients according to their level of acuity as quickly as we safely and reasonably can, and patients can't have it the way they want it when they want it. Apologizing for its being otherwise is counterproductive. I sympathize with patients' frustration, but them's the breaks. If I have ERRED or if the hospital has actually done something numskulled, THEN I apologize (and find out what I can do to make the situation better, etc). Another point: it is not reasonable to expect that every procedure will go according to plan and that the best possible results will be achieved painlessly and without a mess. Apologizing gives the impression that this expectation IS reasonable. Veins blow. IVs infiltrate. Hubs unscrew themselves and get fluid everywhere (usually in CT where contrast material will go everywhere, I find).
Particularly with kids, I'm finding that saying "it's too bad that this happened; we'll try again" works MUCH better than, "Oh, I'm sorry!" Parent anxiety is much lower. My anxiety is much lower. Everyone wins. Sometimes I can't help it, of course. It's hard to resist cuddling a weeping toddler and saying things like, "Oh cutie, I'm sorry your owie hurts so much! Let's go find a TOY!" But I speak of generalities.
As for painful or unpleasant procedures, I have been giving reasonable expectations (eg, "This is going to hurt, but it needs to be done, most people are able to tolerate it well, and it resolves within about 15 minutes for the most part") and then saying I'm going to do everything I can to make it as quick and painless as I can---then I leave it at that. When the "ow! ow!" starts, I say, "This is what I was talking about. I'm working as quickly as I can, and then it'll be over." It just seems to invoke a better overall feeling than "I'm sorry, I'm sorry, I'm sorry." I actually AM sorry, but I don't want my patients to get the feeling that I should be apologizing for doing a really good job caring for them.
FWIW, I adopted this approach from watching the better ER docs with children who have fractures or serious lacerations. They bounce into rooms and boom to children, "I'm going to give you a SHOT! It's going to HURT! You can yell as LOUD as you want as long as you don't move your finger [arm/leg/head]! Here we go!" I noticed that the kids and parents were really philosophical about this.
I can actually grok this now that I think about it. When I go to the dentist and they say, "There's going to be a little pinch," that's crap. I'd rather they give me an accurate representation of the mind-numbing pain that's about to follow and how long I'll have to put up with it.