Don't know if you're still interested in this question, but for anyone who is curious about it, the answer seems to be that there are no pretriangulated categories that are not triangulated.
http://arxiv.org/abs/1506.00887Macioca Pre-triangulated categories are triangulated. Unfortunately, the article is removed by the author with the comment: "There is a serious error at the end of the proof of 3.4 which cannot be fixed. There is a counterexample to the statement of 3.4 due to Canonaco and Kunzer. Lemma 3.5 also needs stronger hypotheses."
