Very interesting and accurate post. I've come across RPs where some characters seemed to be invincible no matter what. It's a turn off or, as Easton put it, a "bubble popper". Now, I won't just repeat what others have said, so here's my two trias: just like nobody is the main character of anything, nobody really is in control of everything. Even players behind an event or casual RP cannot predict nor control all the outcomes, and pretending to choose the one we like best is well... pretencious. We should always leave room for failure because that's the biggest of possibilities most of the times. I sometimes roll the dice when something depends on me RPing a certain ability, giving myself a fair chance of success as well as failure (depending on the experience I have in such situations, among other things), especially when it comes to affecting or reacting to other people's abilities.
And I try to RP abilities I have somehow "acquired" during my PS time (based on what my character has gone through since I started playing and learned in-game). For example, Rangers RP whistles as a means to convey simple messages because it's part of their training. To me it is not enough that a character's background story claims having certain training, a certain object or strength that would represent an advantage over other players if this hasn't actually happened in-game or if they haven't actually spent any time building those abilities through previous RP.
I've been told stuff like "I've always RPed that character has a very strong resitance to poisoned weapons" or worse "My character has the ability to read minds" when that character hasn't really trained any magic way at all (if there's any related to mind-reading for that matter) or learned that stuff from someone else who's bothered to RP their training or research. If one tells me "My character has telepathic abilities so he'll try to read your mind", the key is that they should try. That means they have to attempt it somehow and, most importantly, that they can fail at that. It's not up to them if they succeed. Again, dice could help here or if I'm into the RP they are having I can /tell them what I feel like sharing. But that's me being generous and it should not be taken for granted.
Invincible characters are not necessarily bad RPers, but they are selfish. It's more like they just enjoy roleplaying with themselves... if I have a chance to let them have their fun, I do so by walking away. Play alone, got all you need for your fun. It's tiring having to negociate every little move you make around these characters (especially if the main reason is that they want to be a certain way). This also applies to those who claim to be the most utterly evil, or the most dangerous ones around... and also to those who claim to be the holiest, most delicate and purest spirits in Yliakum.
*Pop* There goes the fun...