I was wondering, from your experiences, how much 2 GB VRAM will be able to handle in a modded Skyrim? (The reason I ask is because my new computer will have 2 GB VRAM, GTX 670)

I don't care about 2048 x 2048 weapons; 1024 x 1024 is enough. What I care most about are landscape textures, and then city and dungeon textures. Will I be able to have 2048 x 2048 landscape textures with a 2 GB VRAM card?

System Memory seems to be a bigger factor for this than the video card. As I have a 2GB video card, and the Hi Res Pack from Bethesda won't run because I have 2GB of system ram (the game will crash after a while).

Note, I have not tried this on my new rig (16GB RAM 2GB Video Card)

If you are using the Bethesda Packs, one of them is for World Objects, one is for Weapons and Armor.

I got about 1800MB and it works fine with 2048 land and 4096 mountains.
Got more lod enabled too.
Shrank my HD dlc to only use 1024 at most.

In my experience, textures regardless of the resolution have never really caused issue, it's only when you throw an ENB into the mix that you start having VRAM issues. Then again, I'm forced to play at 1920x1200 resolution due to my monitor's 16:10 aspect ratio, so my experience may not be typical.