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!
bicycles.stackexchange.com/users/683/whatsisname
My experience from last winter was that I could keep moving through falling sticky snow up to about 6" =
15cm
but it took a great deal of effort,
even more so if your tyres end up cracking through frozen puddles under the snow.
There were sections where I had to pedal hard just to keep moving even down some normally fast downhill sections.
It's great fun -
but maybe don't stray too far from home on your first outing.
bicycles.stackexchange.com/users/825/Duncan McGregor
Depends on how heavy it is and what you want to do.
For a light snow, below the hubs should be fine.
For heavier snows and slush, well, you have to worry more about losing control of the front wheel.
Fatter tires or a mountain bike are easier to ride in slush and snow.
If you want to test the limits go as fast as is safe 2.use
a big gear keep a light touch on the handlebars only brake when going in a straight line and only use the rear brake A fixed gear is easier to control in these situations because of the direct control that you have over the rotation of the rear wheel.
bicycles.stackexchange.com/users/839/David LeBauer
I agree with the other answers, but think one subtle detail is important.
It depends on the type of snow.
Eskimos have 50 words for snow?
Well there really are different types.
Light and fluffy is easy.
Deep and heavy gets to be problematic.
There comes a point, where each forward motion is wasted as the wheel spins, while sinking down through the snow up ahead, till it hits the bottom where it can get traction.
That is very hard riding if it is possible at all.
I am not sure how to discuss the very different types of snow to really answer this question completely.
bicycles.stackexchange.com/users/19/geoffc
If you are talking about riding your bike on normal streets, then this might be helpfull.
I recently got lots of experience riding on various kinds of snow.
Here in Germany, the bicycle paths are cleared and sometimes salt is put on them so that you have various mixtures of mud and snow.
I find it easiest to ride on a road where no salt is put on and where the feet of the people made the snow essentially a flat hard surface.
Riding on that type of snow is possible even with temperatures below 10
Fahrenheit.
If you have snow about 5 inches deep and not yet a hard surface but lots of footsteps and bike traces, then it is difficult to ride.
You bump constantly into tiny snow hills, and you will drive in snake lines or fall off.
If you have salt on the road and it is no colder than 20 Fahrenheit, then it is ok to ride in this snow-water mud.
It get much more difficult to ride if you go below 20 Fahrenheit, then the mud freezes, even if you follow the path of other bikes.
bicycles.stackexchange.com/users/882/Katharina Friedrich
On packed snow (IE from cars), my hybrid performs better than my mountain bike.
The narrower tires cut through and contact solid pavement underneath.
The bigger mountain bike tires can't cut through and end up floating on the surface.
The effect is like hydroplaning.
Very hard to steer, although I don't know the physics behind that.
bicycles.stackexchange.com/users/9074/user9074
I was going to go for an off road ride in the snow this weekend but I chickened out. Partly because it was cold but mainly because I have no experience riding in snow and wasn't sure whether I would get more than 10 metres from my house.  So this got me thinking, if you're riding off road on a path that is virtually untouched, how deep does it have to get to become unrideable?
How deep does snow have to be before it becomes unrideable?
bicycles.stackexchange.com
<mountain-bike><snow><off-road>
bicycles.stackexchange.com/questions/2265
bicycles.stackexchange.com/users/446/Phil Hale
Different types of snow are easier or more difficult to ride a bike in. Fresh snow is easier, tracked snow more difficult. Fresh, powdery snow is easier than wet, solid snow. Moreover, hard ice beneath the snow makes riding difficult. Finaly, the type of bike or tires makes a difference: Fatter, mountain bike tires make riding through snow easier. The answer may depend on your bicycle type, as fatter tires are better for snow. More to the point, it is important to consider the state and volume of the snow. Specifically, fresh snow is not as challenging, even though bikers should beware of icy surfaces, while deep and heavy snow can be problematic. You should also distinguish between fresh and tracked snow, which is more demanding. Finally, temperature is an important factor. Driving below 20 Fahrenheit can be dangerous, even if there is salt on the road.
The answer may depend on your bicycle type, as fatter tires are better for snow. You should distinguish between fresh and tracked snow. The latter is more demanding, even if the snow is not deep. It is important to consider the state and volume of the snow. Specifically, fresh snow is not as challenging, even though bikers should beware of icy surfaces, while deep and heavy snow can be problematic, especially when trying to maintain control of the wheel. Temperature is an important factor, as driving below 20 Fahrenheit can be dangerous, even if there is salt on the road.