While the option of recording your WoW play may not be open to everyone, self-examination is. And it's something we can all afford to do. None of us ever play perfectly, we are not robots. But the key to becoming better is to admit that, to see our failings, and to improve upon them.
Sometimes it's you
Sometimes it is. Sometimes you'll lose an arena or wipe in a raid because of someone else's error -- your tanks messed up the switching, or the person who had a debuff failed to perform correctly. Even in those situations, it's good to think back and wonder whether your performance was optimal. Sure, you weren't the reason why your team wiped or lost, but what more could you have done to perform well on your own personal scale?
Because there's always something you could have done better, and examining that is the very best way to get better. If you allow yourself to assume that you played flawlessly every time, then you're never going to improve. If you don't examine your own performance and look for your errors you may well never spot them. And if you don't know where you're going wrong, how can you get better?
The blame game
This is something that we all do now and then. People do it because it's an easy way out, a way to escape the cold, harsh reality of self-assessment. It's easier to blame someone or something else, than it is to blame yourself. I do it myself, from time to time. I blame gear or my team-mates or the situation or the game, when really I should be looking at my own performance. For clarity, I'm not saying that the reason you lost or wiped is never those things, just that those things never mean you shouldn't also assess yourself.
Let's take a PvE example. The Dragonwrath, Tarecgosa's Rest staff, back in Firelands. My guild decided a destruction warlock should be the one to get it, and he got to the point where he had to head over to the Nexus and kill the dragon within. It was harder for warlocks than some other classes, for sure, but he just couldn't do the fight. At first, he was dying due to the fire that the dragon dropped on the ground, and blaming gear. He felt that with better gear he could live through the fire until we persuaded him that something was wrong and he should move around more. Then it was the adds, then it was the dragon itself. To this day, I don't know if he has killed it. But the point is that, despite being far better geared than other warlocks who'd done it, he was blaming gear for his failings.
And gear is far from the only thing people blame. Bugs, they're a popular one, and yes they definitely exist. But instead of saying "I don't even know what killed me, must have been a bug", look at your combat log and see what it was. Chances are, while glitches happen, it was you.
Sucking at something is step one
While that last section may have come off a bit harsh, it's really my way of trying to say that we blame things for our own failings. It's human nature to do so, but admitting your mistakes to yourself, and to others, is really important to improving your gaming performance. I'm not saying you should spiral into a pit of self-loathing every time you lose an arena, but that you should look at what you could have done better.
So I'm learning. I've got some friends who don't mind having a terrible ele holding them back. We're doing OK, we're winning matches, we're not remotely high-rated but we're all learning. And it's fun, we have a laugh, we giggle at our failings and discuss how we could have done better. As we learn from our errors, we're getting better. Soon we'll go from being kind of OK to being pretty alright, and then it's a slow climb to actually quite good.
As I said above, it's not the case that it's never someone else's fault. And if your team-mate or team is regularly having trouble with the same thing, what should you do? Have a raider or PvPer who never uses their personal survival cooldowns? How do you tell them, constructively?
First up, be kind. Imagine it was you who was messing up, what would you want to hear? The answer probably isn't "OMFG NOOB stop not using your personal cooldowns you idiot". Instead, try something like "hey [player], could you try using [ability] when you're getting low on health?" Or even more subtly, "man, I suck at remembering to use my personal cooldowns when I get low" could be good as an initial approach.
Likewise, being that guy in battlegrounds who says "OMFG SCRUBS L2P" isn't doing anything useful for anyone. In fact, that guy is a big part of the problem. That guy probably ends up losing a lot of the BGs he's in because he's so convinced he's doing everything perfectly and everyone else is awful. If you know a good strategy, share it. If you see that several people are making mistakes maybe say something like "we don't need seven people at Lighthouse, if we all push Waterworks together we could take it!" Don't be that negative guy, stop blaming your team every time, think about what you could have done better.
Humility is a great virtue. You'll learn a lot more by being humble, asking for advice from more experienced players, admitting your flaws and failings, than you will by refusing to see your mistakes. As an aside, if you want to start recording your play, and have the capacity to do so, OBS, DXTory or FRAPS are great pieces of software. We all suck sometimes, and it's OK.