My team just took tons of heat because there was a complex performance issue that needed our full attention. We clearly communicated that we would need a large chunk of time to solve it, and while we tried to stay responsive, a lot of stuff had to just get ignored for a while to be able to make any progress on it.
We're finished now, and what we did will eventually save the client literally months of time over the long run, but because we weren't constantly placating the people who wanted to see the job "getting done," we put ourselves at risk of losing the contract.
All's well that ends well, I guess. We finished the task, and now we're back to churning through the high visibility low-effort issues and everyone's happy again, but all of those small issues combined don't have as much impact as the big task we just finished.