Ok, a lot of other things (shows/books/movies, w/e) have different views on time traveling than CT does. For instance, in Back to the Future the main character (it's been a long time since I've seen it) has to try to avoid changing the past or parts of his body disappear (or so I remember), anyway, that's obviously different than CT.
I don't mean this as a flame to anybody, but all this "TTI works for people who have a strong connection to the time period.." or "Who get's TTI?" is crap (I didn't know how to put it more politely <_<). Anyone who travels back in time 'has' TTI. This is one of the reasons that I hate giving it a name, if you have something called Time-Traveler's Immunity, then you're going to want to know how someone can get this immunity. However, the truth is, at least in Chrono Trigger, that it's an innate law of time-travelling that traveling into the past and effecting things in the past won't make you disappear or anything. In CT it's impossible for something's existance in the past to affect its ability to exist. Essentially, when would it disappear?
Say I go back in time 20 years. I'm intent on killing my father so that I won't be born. If there were no such thing as TTI, when exactly would I go 'poof!'? Would I disappear once I kill my father? Would I disappear once I pick up the knife? Would I disappear once I get the intent to kill my father? Even if it's a minute before I kill him, he's going to die, and I'm not going to be born. But if I go 'poof!' then, then my existance won't be in question since I won't have killed my father.
This is hard to explain, but from the timeline's point of view my existance is no less in question after I kill him then it is once I appear 20 years in the past.
This is all in theory, of course. I <3 my dad, and I don't have a time machine that can take me 20 years into the past, in case you didn't know.