I've got four pieces of armor on, each -28% destruction cost. I've got two swords: one with absorb health and fire damage. The second with absorb stamina and frost damage. My research showed that all four weapon enchantments were of the Destruction school, and therefore would have infinite charges. After a few hours of questing and dungeons, they both have 1/2 to 3/4 of their gray charges bar left.

Did I do this wrong or misunderstand something?

Thanks,
Bohiti

From my understanding, your skill in destruction allows you to have more uses out of a weapon's charge meter.

Having an enchantment that reduces destruction spell cost doesn't effect weapons because they're not spells that you're casting. If you were able to increase your destruction skill, not a casting cost reduction that would be a different story.

Enchanted fortify destruction items definitely increase your weapons use just as high destruction does. I am certain of that from personal experience.
However, I have never had 100% cost reduction so I'm not sure what happens at that level. Possibly there is a wrap around problem over 100.

primetime bump? Anyone heard of wraparound bug for destruction spell cost reduction affecting weapon charges?

Which platform are you on? From Wikia :
Fortify school enchantments do not interact with weapon enchantments in the same way across all platforms. On the PC weapons appear to always drain charges when used. On the PS3 and 360 100% or more fortification in any school will reduce the charge cost of a weapon enchantment to zero. It is unknown which situation is a bug and which is the intended mechanic.

PC. Doh. Thanks for the information. This would seem to be a glaring inconsistency. I guess we'll find out which is right with the next patch.

I would guess though that the intended mechanic would be that the number of charges would not be effected as you're not really casting them, just using them.

spell cost and weapon charge aren't the same thing, that counts only for spells, not weapons