I think those people who state that they are busy are not really busy. They are simply trying to “avoid facing the real problem(s).” There might be no denial that these types of people are, “Overloaded with work.” Undoubtedly, they have plenty of work at their disposal to stand daily. A simple suggestion: hire more people instead (to help you get done with your work).
Furthermore, they should not convey, a negative vibes among their colleagues and friends of stating or being, “Busy” (You may wonder, why negative vibe?). Stating or being too busy is somewhat negative vibe. Not only this habit shows that you are being ignorant and overloaded with tons and tons of work but also gives a negative impression of your abilities and capabilities (things that you can do as a human species, possibilities, ‘khubility’).
There are smarter and intelligent way to deal with this type of behavior. Similar to solving a math problem, one could simply say, “I am really working hard on this issue (step), I will come back at that (step) later” or, “I have some very important things happening right now...” etc.
Prioritizing the task will ease out the common trouble between being busy and not-so-busy. If you don’t know how to prioritize the task, ask in your inner self, perform cost-benefit analysis, do simple soul searching on task or tasks. What is important is NOT how much effort you put into solving a problem but how many problems you can solve in limited time. Because our time is limited and we are not here forever. You will not be judged by how much workaholic you are/were, but how much collaborative and supportive you can be, also by examining vast and wide range of problem(s) in parallel. This is also called in plain simple English, productivity. The more productive you are [with people], not with task(s), the better you as a person become. Because it’s the human connection that matters, everything else is secondary.