Ranma is sometimes portrayed as a braggart in fan fiction; using statements such as I am the best (which incidentally I don't really like very much and find annoying when characters say that - for example Wolverine from Marvel comics saying he is the best at what he does and what he does is not very nice) and saying "Ranma Saotome doesn't lose". I can't say conclusively that Ranma has never said things of that nature in the manga, only that I do not recall him doing so (I'm almost positive he does not at least for his martial arts abilities). I do not see Ranma as a braggart; using the definition very boastful and talkative person for braggart and speaking of yourself in superlatives for boastful.
There were several instances of Ranma downplaying his skills that I can recall such as during the first volume when he (technically she) downplayed his abilities and during gym class where he did so yet again. This by Japanese culture is the proper response and it also has an added advantage for Ranma in that it leads his opponents to underestimating him. That really is not the reason though. The main reason I don't think Ranma did so is because Ranma is a very secretive person who hates talking about himself and did not talk about any of his previous battles (most things about Ranma's past we learn second hand), Akane even complains that Ranma never really tells her anything in the manga. If Ranma boasts about say beating Saffron (commonly commenting about beating a god) it would be extremely out of character, since it would be out of character for Ranma to even mention fighting him in most situations.
In the event that Ranma did once say something of that nature in the manga, I'd like to point out that the manga occasionally introduces a trait or piece of information that conflicts with other arcs; such as Ranma developing an inability to play cards which is countered by multiple other arcs in which he can. When there is a conflicting trait, I tend to discount that trait in favor of the one that has more supporting evidence as being my personal canon. This type of error is common in many works, particularly in long works written by many authors like Batman. As for the Ranma manga, it is a comedy so it uses the rule of funny, i.e. being funny is more important than being consistent. I mostly go by the manga, since I like it more than the anime and the majority of fan fiction, but I'd also like to point out that the anime and fan fiction tend to change or exaggerate character traits (not that the manga is entirely free of doing this as well) so it is also possible that people view Ranma as a braggart from the anime or just from other fan fics.
Of course it is also possible, as pointed out to me on a forum, that it stems not from anything Ranma ever said but from some of Ranma's character traits. Such as his competitive nature, dislike of losing, and belittling/underestimating other characters skills.