First: a clearer line between "being a dick" and "being a bully" needs to be drawn for this sort of discussion. Being a dick is things like calling people fat, or dumb, or retarded. Being a "bully" is a different beat entirely. Sending texts or posting on Facebook or whatnot about someone, completely unsolicited, for no reason. Harassing them over whatever the topic happens to be. Etc etc. It's more of a chronic "I am regularly victimizing you, and doing it as much as I can" than it is "you're fat, trolololololol". Petty insults and whatnot, yeah, people do need to learn to deal with it. But people constantly assailing you with messages about your own worthlessness? That's crossing a goddamn line, unless the person being victimized is, like, Hitler or someone.
I'm gonna let loose a bit. The mentality of "just deal with it" is a goddamn immature attitude. It's shifting blame and responsibility onto the person being victimized--because clearly, they should have "just dealt with it" and not let it affect them. Aside from this being, on a basic and abstract level, the same thing as blaming rape victims for being raped (i.e., both shift the blame to the person being targeted, and away from the person doing the targeting), it's just plain stupid. In the real world, if someone makes an inappropriate comment in the workplace, they can be fired. If someone keeps harassing you with messages through Facebook about how you're "a fat stupid *****" or "a worthless moron who should just kill themself," in the real world, you report that shit to the police and they (hopefully) give the person a visit. The point is that in "the real world" the same behaviors that bullies often partake in are absolutely not tolerated. So if the argument is that people need to prepare for the real world, why would they need to learn to deal with it all, stone-faced, and just not let it get to them? It's honestly better preparation, I think, to teach them to report the offenses to the relevant authorities.
Also, telling people to "just deal with it" might work well and fine for well-adjusted adults or unusually mature middle/high schoolers. But the truth of the matter is that people in middle school and high school and whatnot are not the pillars of rational thought and self-confidence that the comment implies they should be. Frankly, people at that age--which was me but a few years ago--have a very, very hard time just shutting out what other people say and think. Especially when the harassment often takes the form of attempting to invade someone's social and private life with utter filth.
So, to put it short, I think the idea that people should just "suck it up and deal with" bullying is absolute garbage. It seems to rely on a false assumption that people--and let's remember the age bracket typically in question with bullying--even can do that, because it's oh so much more complex than "you just do it". It shifts blame and responsibility to the people who are being hurt by this, and away from the people who are so often actively seeking out ways to cause someone pain. It's unrealistic in preparation for "the real world"--that sort of shit is not tolerated in any sensible real world situation, so why should it be tolerated amongst the younger age brackets who can be most negatively affected by it? Why should they be expected to already have thick skin, when they barely know anything about the world or about people in general?
This is all not to say that people shouldn't develop some degree of thick-skinnedness. There are tons of examples of petty bullshit people need to be able to deal with, and it sounds like that might have been what OP was talking about--a few insults here or there, general non-targeted dickery, etc. If you lose it every time someone calls you a name, that might be (depending on the circumstances) something to look at changing. But "bullying" as I understand the term goes beyond that, into the territory of harassment and continued, serious emotional and verbal assault/abuse (and perhaps physical, as well). Shrugging your shoulders and saying "it'll always be there" or whatnot and telling people to just deal with it is an infuriating way of addressing the situation, because it misses the whole damn nature of the situation in the first place.