Would be a neat idea, as long as it could be toggled. However in heavily populated areas (Like Ul'dah or spots where more than a certain amount of players appear all at once.) it should be deactivated by default. Can you imagine trying to make your way around and all you see is 20+ chat bubbles all appearing at once in an area? You'd have to disable them, which entirely defeats the purpose. The Phantasy Star series (PSO/PSU) had a feature like this, where only a certain amount of chat bubbles could appear on-screen at a time. Each new one would remove the other one.

But there's also a really big fault to having chat bubbles with the length of how much you can type at a time. For somebody who types out a HUUUGE wall of text (I'm guilty of this when sending /tells sometimes.) unless it automatically shortens it you'll have a giant text bubble clogging up your screen. Worse if somebody decides to be a douche and spam. It can, and will happen.