I'm perennially swamped. It's interesting to see how students react to it.
Some go off and choose other advisors, because they feel they want or need have more one-on-one attention and face time. I wish them well, generally. One hopes they find what they are looking for and in any case are not embittered. I believe the Wiccan expression is "merry meet, merry part, and merry meet again".
Some hover in my doorway, hesitant to come in, and ask "are you busy?". (The answer is that I am always busy, but I would not take someone on as an advisee if I wasn't prepared to spend time with him or her.)
Some barge right in, even when other students are sitting in my office, or jiggle the handle even when the door is closed and locked and they have already knocked. (This last one is a pet peeve of mine, especially because it's disturbingly loud from inside the office.)
How many of you have been "grabbed in the hall" as a teaching assistant or instructor?
I don't mean literally seized, but rather "tapped on the shoulder" or just called as you are walking somewhere (when it's not your class time or office hours). How did you respond?
Perhaps your students respect your personal space. Nowadays, students approach me anywhere, any time, and ask me to give them a moment's attention, which I always do, unless it's going to make me late for class. (It's certainly made me late for everything else, though!)
Back when I was a TA for CS 125, Introduction to Computer Science, at the University of Illinois, students used to literally grab me from behind. One time, I shrugged it off reflexively, and the student recoiled, realizing it was something he shouldn't do. I said as much. Another time, a (male) student followed me into the men's washroom to ask a question, and I had to explain that this was not quite fair to some of the students.
Maybe I'm not being emphatic enough...