I took a job with an employer in London who didn't get my taxes right, but I was none the wiser of course. A few months in, my tax increased by a significant amount from one paycheck to the next, and this tipped me off to something being wrong. I talked with my employer who looked in to it, but this kept on being the same for about three months. At this point, I'd been on the job for maybe 7 months or so. Finally I'm told I have to sort this out with HMRC myself, so I called them. They were very helpful and pleasant to deal with, and the person on the other end of the phone call very briefly looked over my account and determined I've been overpaying every single month – by a lot. They noted as well that HMRC had indeed changed my tax code to an "emergency rate" or some such about three months prior to the call, but failed to note the wrong tax code that was already there in the first place. This person did sort it all out however, and a couple of weeks later I got a check in the mail for over £6000.
It did finally work out in the end, but I'm not sure I would've noticed if they hadn't hiked my tax so significantly, and I don't know what would've happened then since apparently they hadn't registered the error either.
Moral of the story: HMRC are there to help, but you may want to be proactive and call them if you think anything might be wrong, or you want to double check something. Their phone service is very helpful, and usually not too busy.