This is a thing in Guildwars 2, you can double tap movement to roll/dodge in a direction and be immune to damage during the roll. In that game's case, it does not take very long to recharge, but then again there are many class mechanics that are tied to the dodge mechanic in some way. Also, there are both 3D damage zones that can be avoided without dodging entirely (you can jump over a wave of energy for example).

It's makes for a very fun game that feels more like I'm playing Smash Brothers than it does a typical MMORPG.

The problem...when you play GW2 and come back to FFXIV, you immediately learn that FFXIV's engine is just too laggy for it. Like, what is happening on my screen just does not match what's going on in the server. If I cast a spell, I can move at like 85% cast bar. The actual monster graphical representation of their attack has no meaning for timing (as in, the orange damage indicator is sketchy; you might take damage even though you have absolutely gotten out of the damage zone... nevermind the monsters actual attack animation).

In the same vein, playing with my wife, if we have a foot race in a random zone, I will be in front of her on my PC and she in front on hers.

Full disclosure, we live in Japan on Japanese server and have a very fast connection/low ping.

I can only see something like this work with mechanics like Shelltron or Third Eye (as mentioned above).