How fast can I deal damage with a weapon?

All enemies (and I believe this is actually Entities, not just hostile mobs) have an invulnerability period after you strike them. That is when they turn red and only lasts for a split second (for the most part).

You can see it when you are striking a cow or the like and while its harder to see on things like a blaze, you can swing your sword to check. A blaze I believe takes three hits with an Iron sword to kill it. If you swing your sword fast you will be able to hit it I believe 4-5 times before it dies as only three of the hits will actually register and do damage.

Please note that this is regardless of how fast you swing, and in fact the faster you swing the more wasted uses of your weapon will occur so if you are buffed by a potion, beacon or whatever else may be available to you, then watch your attacks a bit more carefully.

Side Note: A mod where this is very easy to see is Aether and their cube boss (Which is what first brought it to my attention). You could go through 2-3 picks constantly hacking at the boss but any strike done while it was red would not actually cause it damage when it would only take a pick and a half or so of hits to take it out.