While big, flashy encounter mechanics are a part of this problem, there are other elements that contribute to it. The press of bodies in melee range of a boss often creates a blindness - players simply can't see anything around the players in range of them. This gets even worse when spells with distinctive visuals are going off at the same time - if the DPS warrior hits Thunder Clap and then Bladestorm while the ret paladin is hitting Divine Storm and they're all packed inside an big glowing yellow or purple bubble while warlock green fire rains down from the sky, it's not surprising if something is not immediately observed. There's so much going on. Add into that a few boss mechanics, a healing rain or two, adds - things get more and more complicated and details become easier to miss.
Emphasizing personal visuals and toning down the stuff others cast is a step in the right direction. We'd still have to worry about boss mechanics, of course, but at least when stuck in the murder pile with all the other DPS we could have an easier time of seeing things that simply have to be seen and reacted to as quickly as possible.
Another step might be to have certain visuals be outright invisible to anyone else. One example is the big floating black skull that appears over a warrior's head when he or she enrages. That might not need to show up on anyone else's screen apart from my own. The danger there, of course, is that things that make the world feel cool and lived in together might get lost.
It's worth keeping in mind however that a lot of this is raid mechanics, however. The trash before Sha of Pride comes to mind as just ridiculously full of effects, and perhaps too many of them - we lose one or two people to their computers locking up on every trash clear before a Sha pull. It's also a problem when the mechanics have big, flashy visuals that players need to get out of, except getting out of them won't save you always because the actual effect is larger or smaller than the graphic chosen to visually display it. It's one thing to have a boss do a gigantic heroic leap/shockwave combination that spends spokes of fire out but if those spokes of fire aren't even where they appear to be on the screen, it becomes an exercise in pure frustration.
This is definitely an aspect of raiding that needs to be toned down, especially with LFR and all raids save Mythic being flexible in Warlords of Draenor. Hopefully it can be done without losing the coolness factor of seeing spells and so on flinging around a room.