The item states that it has a 3 second cool-down in order to increase your maximum mana by 4. This cool-down is not happening. I can cast 3 spells within 3 seconds and get 12 mana. Is this a glitch or am I just misinterpreting this item?

This has been possible for a long time (Tested this ages ago in early Season 2). I think internally the item actually still has a reservoir system (unlike other items that trigger on spell casts which have had such mechanics removed).

The reservoir system for ToG is relatively harmless.
The reservoir system for Sheen was not. Extra 300 damage burst for 1200 gold early game when champion health values are low and noone has purchased any extra survivability.