The biggest problem is the downtime for other players. In a group of four or more players, with complex characters in a system that allows for loads of options, any player who uses their turn to deliberate will enlarge the downtime for others. I've not yet read a game system that forbids this, meaning that the system allows for long, deliberate turns that make other players snooze or reach for books and other distractions.
Once, I heard an actual play recording of Escape Or Die! by Fred Hicks. The game includes a mechanism to help with this problem. Play moved around the circle, with each player framing a scene for the character on the left. In the background, a timer was counting down a number of minutes (one less than the number of remaining characters). If the timer expired, Doom was increased by one. The only way to reset the timer was to make a complete lap around the circle of players. Listening to the game, I enjoyed the tension of the doom clock and the chorus of "Doooooom!" when it reached zero. I also remember one player who liked to narrate long, drawn-out scenes and actions - oh the frustration! His penchant for elaboration chewed into that timer, leaving less and less time for others to play, but at the same time engaging the players because they knew they'd have less time to frame a scene and act it out.
Another choice is a game I've neither played nor read: Sons of Liberty. From what I can glean here and there, there are no turns, in the sense that I've discussed above. If you have cards in your hand that let you take an action, you can play your turn. If anyone spots an actual play recording of this, let me know. I can only imagine the potential chaos.
Board games typically make use of a turn sequence. Monotony, er... Monopoly is the classic example. It not only has a lengthening downtime, it also has a horribly demotivating death spiral. I don't play it for those two reasons. It's rarely fun for everyone.
If you're wondering about the significance of this on your favourite games, and perhaps thinking that I'm spilling a lot of pixels for no real benefit, consider changing the turn sequence mechanism of chess. Chess has alternating turns in which each player is allowed to move a single piece (complex moves like Castling notwithstanding). Now suppose you were to play chess with the same turn structure as Warhammer. Players take it in turns but on each turn can move every piece once. What does that do to your chess strategy? 
Perhaps take it further and apply a Sons of Liberty approach. You can move one piece at a time, but you can keep moving pieces as fast as you can move your hand from a finished move to another piece. What does that do to your chess game?
The mechanism for turn sequence has far ranging effects on the game. It's the framework for the framework, so to speak. In thinking about my own game-in-development, this is a key area I've yet to decide upon. At the moment I'm ruminating over the Escape Or Die! concept and a highly procedural concept (e.g., A Penny for my Thoughts). We'll just have to see how it goes.
1. Conversely, what would it do to your Warhammer strategy if you could move only one piece or squad in your turn?