This could be a very technical discussion, so I will try and keep it simple. The answer to the question of can it be too cold to snow is no, but is gets increasingly difficult the colder it gets.
Let’s familiarize ourselves with an important concept. Warm air can evaporate a lot more water than colder air. There is more water vapor present for producing precipitation in warm air than in cold air. When any air mass reaches the point where it can’t absorb any more vapor, it becomes saturated, produces clouds and finally precipitation.
As the clouds form, water droplets start to condense out of the air. As more droplets form, the greater the chance of them colliding and forming bigger drops. When they get big enough, they fall to the ground as precipitation and if the air is cold enough, they fall as snow. But when the air is really cold, it can’t evaporate as much liquid water to form water vapor, and therefore has less precipitate content for making snow.
As the air gets colder there is less water vapor present. The less water vapor in the air, the less precipitation that can be produced. When it is 32°F (0°C) there is a lot more water vapor present than at 0°F (-18°C), so a lot more snow can be produced. The snowflakes are also usually bigger.
However, when the air gets really cold, the snowflake decrease in size, there is less precipitation, and it takes a lot longer for the accumulations to occur. Therefore, as it gets colder it takes much longer for the snow to accumulate on the ground than if it were a little warmer. And when it’s really cold, say below -40°F very little precipitation can be produced. Ice crystals can form, but they are extremely small.
There are a couple other reason why it’s hard for snow to form in really cold temperatures. One of them being, there is less evaporation taking place. For this reason, cold air is usually much drier than its dew point, unless you happen to live next to a large body of water that hasn’t frozen over and is warmer than the ambient air temperature (Lake Effect Snow). The drier the air, the colder it has to get to reach its dew point and ultimately before condensation will occur.
The other reason snow has a hard time forming in really cold temperatures is that the atmosphere is usually a lot more stable. It’s hard to get really cold, dense air to rise, expand and cool so that the little amount of water vapor will condense and form snow. There are exceptions. One of them being a mountain which can force the air upwards causing expansive cooling at which point the air will achieve its dew point. Another being a warm air mass moving over top of a very cold air mass.
So putting this all together, snow can form at any temperature provided the right conditions exist, but it has an increasingly difficult time forming in really cold air because there is less water vapor present and the atmosphere is more stable.
Right around freezing is where you will see the most snow. This is because the air can absorb more moisture before the dew point is reached, and yet still be cold enough to freeze. Once it drops below -20°F, your chance of snow drops quickly. Ironically, when temperatures are less than -40°F, snow can form without ice nuclei. However, these ice crystals are extremely small and accumulations are virtually unnoticeable. Bottom line is, it cannot be too cold to snow.
Where I’m at in Colorado, I’ve seen it “snow” at about -5°F. I put snow in quotation marks because the snow wasn’t composed of the typical looking snowflakes. Instead, itty-bitty ice crystals that looked like shards of glass were drifting with the wind. I could only see them when looking at the porch light.