Today I recently had this issue arise.
I've never had this issue before, so I can't say what caused it.
However, according to some other forum posts relating to this issue, I skimmed through several of them and none had a straight answer detailing how to deal with this problem.
I heard the word router thrown around a couple times, and I'm thinking my router company may be the issue, although I can't confirm or deny this.
My linkshell buddy also just recently had this problem and managed to fix it, but he's not going to be online until later, and even then, I'm going away to a Legend of Zelda concert this weekend, so I doubt I'll get to talk to him one on one.
And to add on top of that, I can't say 100% for sure that he had this very same, exact problem as I am. It just sounds very familiar.
I would also like to add that the rest of my internet connection (Firefox, Skype etc.) is working just fine and normal.
My game was playing and working just fine also up until I woke up today to sign in. I had a few random D/Cs here and there while crafting lately, but that is about it. And that is pretty normal for most FF XIV players I know anyway.
So, if anyone out there can assist me, or has the answer from past experiences of this very sort, I would be greatly appreciative. Thank you~!!