There are those that claim you can't go back in your own timeline and change anything without introducing a paradox of some kind, correct?
First of all there is the case of the Ignorant Time Traveller. If someone who had never heard of the Titanic were to go back in time and inadvertantly convince it to go off-course (and therefore avoid the iceberg), there is no paradox, since there is no reason why the Ignorant Time Traveller wouldn't do it again.
The Ignorant Time Traveller could travel to places and times with which he or she was unfamiliar, and even if the changes were deliberate rather than accidental, they would still not necessarily introduce paradoxes.
It is naturally possible that the Ignorant Time Traveller could accidentally kill a key ancestor, or somehow affect something that prevents his or her ultimate ability to travel in time, but it's not guaranteed - it is possible that the Ignorant Time Traveller would not, and that's the point.
There is also a scenario where time travel is possible even with those who aren't ignorant. Though the Knowledgeable Time Traveller could not prevent the Titanic disaster, he or she could arrange for a rescue ship to save everyone and take them somewhere (or some time) where they couldn't (or wouldn't) tell anyone who they were. Perhaps even to a point in the future, beyond the point from which the Knowledgeable Time Traveller originally went back.
Of course, this would require a great deal of precise knowledge about the Titanic disaster so the right evidence could be left behind. Perhaps he or she would have to bring and leave behind certain realistic-looking corpses, know exactly when and where to show up, and with what, in order to leave no evidence of the rescue mission behind. In essence, he or she would have to be a Very Knowledgeable Time Traveller to leave no evidence of the incursion.
Though paradoxes are still possible, the absence of paradoxes is also possible.
Therefore, time travel within one's own timeline without paradox is possible at opposite ends of the spectrum: the Ignorant Time Traveller and the Very Knowledgeable Time Traveller, but not necessarily at any point in between.