The more detailed answer is how DXDiag registers memory.
Technically speaking you (and I) have 4GB of VRAM modules on our cards, realistically, this means you have 4 usable GB of memory.
DXDiag however, also includes pagefile into the count. You can process up to (in this case) 8GB of memory for the card, but only 4GB (the amount on the card) is actually fast enough to be considered usable. Hence if you ever go over Vram limits, it stutters because the system is waiting on the much slower memory outside of the card.