Disclaimer: there will be spoilers in this post. Also, FFVI was my first FF and it's in my top 5, so I'll be objective.
The main problem I have with FFVI is its excessively large cast. Half of it is made of characters who have no plot relevance and the other half suffers because of them.
Let's take Terra and forget for a minute the brutal treatment of her character in Dissidia. She's actually a pretty damn good character in WoB. She's cute, she's a competent fighter, she has just enough angst to emphatize her situation without being annoying and she gets just enough screentime to make the player get attached to her.
The WoR happens.
The result is this: http://tvtropes.org/pmwiki/pmwiki.php/Main/Chickification
This is where Terra's character gets utterly destroyed. This is also why in Dissidia she's a spineless wimp who needs male support (from Onion Knight, Cloud and Vaan, just to humiliate her even more) to be able to stand on her feet.
Thankfully she eventually snaps from her pitiful situation thanks to the power of love and... and that's it. Her character development and plot relevance stop there until the very end of the game.
Similiar arguments can be made for the other characters who stop having any real plot relevance in WoR, such as Sabin.
Then we have a bunch of filler characters who really don't have any reason to be there. Gau? Umaro? Gogo? Relm?
Speaking of Relm, she's the quintessential satellite character: http://tvtropes.org/pmwiki/pmwiki.php/Main/SatelliteCharacter Take Strago and Shadow out of the picture and her existance becomes entirely meaningless. The funny thing is that both Strago and Shadow have no real plot relevance either! Strago is a simple Mr. Exposition character at a specific point of the story (http://tvtropes.org/pmwiki/pmwiki.php/Main/MrExposition), while Shadow barely does anything other than pissing Kefka off at the end of WoB.
This is where FFVI's narrative problems become obvious: it relies on very, very weak connections between non-essential characters in order to make the plot move forward.
Celes and Locke seem to be the only characters who have consistent good writing for the entire game. In my eyes they're the real protagonists.
This game is exactly why I'm wary of RPGs with large playable casts.