How deep does snow have to be before it becomes unrideable?
Riding in snow depends on a bunch of factors, so you won't really ever be able to know unless you just go for it and try it out.


What bike/wheels you have


Should be obvious. Mountain vs Road, 26" vs 29", wide vs skinny tires, etc


Your skill level in snow


The better and more comfortable you are in the snow is important. I'd say its the most important factor apart from the bike. While some have trouble steering in just an inches, others can handle nearly a foot or more.


Tracks in the snow


You mentioned fresh snow, but I'm going to include this as part of a general answer. The difference between fresh snow and tracked snow is huge. Tracked snow is vastly more difficult to bike in. With just a few inches of solid snow and some criss-crossing tire tracks, it can be nearly impossible to ride at any significant speed.


Consistency of snow


Lastly, fresh powdery snow is a lot easier to go through than wet solid stuff. But powdery can be far worse if a hard icy surface lurks below the snow.

Bottom line: Just go out and ride through snow and see what you can handle, then ride through something a little tougher. Best part is, the soft snow makes falling easier!