I think one problem many people have when trying to move up is they dont "win" fast enough, or cant build whatever they want too whenever they wish.
Higher difficulties are annoying that way. You lose races to techs and wonders you're used to winning. You arent always in the lead, or very near it. Your military seems anemic compared to the way you're "used to" running it. Etc etc etc.
One of the key factors of higher difficulties is that it SHOULD take you longer to equalize and ultimately win. If you get too set in your ways, teching and building in a pre-set fashion that you have "perfected" on a lower difficulty, then you will indeed struggle to move up.
What I am trying to say is that you need to make strategy adjustments as you go up in difficulty, because what is a "winning move" on one level (like the CS slingshot, thats just insanely overpowering) is next to impossible even just one level higher.
My big hangup going to Emperor and Immortal is that it seems to take SO LONG, into Modern Era even, to begin to feel like I have the game equalized, much less "in hand". When I first tried BTS, I played a game at Noble, thinking I needed to learn the ropes of the new leaders, tech tree, buildings, wonders, and features, etc, but it was a joke, I learned nothing, in fact, I learned the WRONG things, in some ways.
My best advice when trying higher difficulties is HANG IN THERE! You would be SHOCKED how easily it is to turn a bad position around, and even more shocked at how strong you're position really is, even if you think its "weak" or "lost". Some keys to my success at higher levels were "re-learning" how to do ordinary things like chopping and whipping, or changing civics/religions. Timing is critical at these levels. Knowing how to use whip/chop overflow to the best advantage, knowing which wonders/techs to skip and which to beeline, thats how you climb the difficulty ladder.