There's something since the 2.1 patch where the actual gearset name doesn't update like it used to. I played a Lancer from 20 to the 40s while constantly changing out my weapon (what the gearset name was originally based off of) and my Gearset name remained LNC (Lv. 20) or whatever. Eventually I just renamed the gearset names myself to just reflect the job in which the gearset is for.

If your item level isn't increasing after getting substantially better gear (it's displayed in two spots in the char screen) then that's an issue. Minor upgrades don't seem to make an impact though. For example, I upgraded my Bard out of the Darklight Gloves for the Ballad Gloves out of CT and my ilvl didn't budge at all.