I think people seem to be mixing up the definition 'immortal' with 'immune' these days, or at least think 'immune' always has to be a trait. It's not. In fact, I've roleplayed with someone whom had an immortal character, but even said in character that it didn't mean she was immune; And thus, she could drown, or die of anything, really. I myself play a somewhat immortal character, (Though may die due to a permanent injury in his head some day) and really, I wouldn't consider him close to immune.
More likely hard to be killed when it comes to injuries, often attempting to flee before it gets too bad for him, or tries to stop the fight with words sometimes, even. By knowing he can't exactly die, (Though, which isn't always involved in immortality) and being so emotional sometimes he's always at least remotely depressed if not really from past events. In fact, he's even went a little bit insane because of it!
And in case yes, I am talking about Exxy. BI *Pushes haters away*
But really, I all find it a little bit silly if you don't seem to at least put flaws in being immortal. In fact, sometimes living for so long would be a pretty big disadvantage if you think about it.
Also, I apologize if I seemed to mix up some definitions; I'm merely stating what I've been taught and giving off my own opinion, really.