I'll bet you didn't know this
Can it be too cold to snow? People from warm climates might be forgiven for thinking that's a crazy question. After all, it only snows when it's cold, so the colder it is, the snowier it must be. Right? Wrong. Arctic climates often get surprisingly little snow. Barrow, Alaska, for example, gets less snow than Chicago in an average year, despite having winters that average 39°F (22°C) colder. So does that mean it can be too cold to snow? Well, people from cold climates might be forgiven for thinking it can, since they have lived through a lot of cold winters and may have noticed that the coldest weather of any given year has never been associated with snow. That isn't really because it's too cold to snow, but because it's too dry. (The coldest weather is almost always associated with very high pressure and very dry air.) The truth is that it can never be too cold for snow, barring a drop in temperature all the way to absolute zero (-460°F or -273°C), in which case snow or lack thereof should be the least of your concerns. But even at balmier temperatures than absolute zero, below, say, -20°F (-29°C), it can be too cold for a lot of snow to fall.
Let's consider an illustration of why cold doesn't always mean snowy. Balmy Nashville in Tennessee (where the cotton blooms and blows), gets more snow in an average year than the frigid South Pole. Nashville gets about 11 inches a year and the South Pole gets between two and nine (my sources vary, perhaps because accumulations can be hard to measure there with the extreme blowing and drifting). So why isn't there a two-mile-thick glacier covering the Grand Ole Opry? As great a boon to civilization as that might be, snow in Tennessee melts. "But where are the snows of yesteryear?" Villon asked. In the case of the South Pole, the snows of yesteryear are just now settling in and making themselves comfortable. A few inches a year may not sound like much, but when it doesn't melt it really starts to add up. So far it sums to 9,000 feet, the depth of ice at the nether pole.
So obviously you need more than just cold weather to make snow. The other thing you need is water vapor. Nashville has plenty of that, thanks to its relative proximity to the warm Gulf of Mexico (500 miles to the south) and the prevailing westerly winds that bring moist air all the way from the warm Pacific. The South Pole has none of these factors. It is farther from open water, the water around Antarctica doesn't evaporate as readily because it's so cold, and the prevailing winds tend to carry what moisture there is parallel to the coast, and not toward the interior. Barrow is in pretty much the same boat as the South Pole, but to a lesser degree. It gets several times more snow, and summers there are warm enough that the snow melts every year.
Another thing to consider is that frigid air can "hold" less water vapor than merely chilly air. I put "hold" in quotation marks because the air doesn't really hold or support the water vapor in any real sense. The nitrogen and oxygen coexist with the water vapor in the same volume without interacting with each other very much. It is the temperature of the volume, not the presence of other gases, that determines the saturation vapor pressure of water. Saying the air is "saturated" is another way of saying the dew point has been reached, or that the relative humidity is 100%. "Saturation vapor pressure" is also a rather misleading term, but it hasn't yet been overtaken by the arguably more appropriate "equilibrium vapor pressure." For most practical purposes, saturation can be thought of as the greatest amount of water vapor that can exist in a volume at a given temperature without some of it becoming liquid or solid, but see below for important exceptions. The higher the temperature, the higher the saturation vapor pressure. If you could turn all the water vapor in a volume of saturated frigid air into snow, you would get less snow than if you did the same thing to a volume of saturated chilly air. How much less? A lot less. At 32°F (0°C), a cubic meter of saturated air (strictly speaking, saturated with respect to ice) contains about 2.7 grams of water vapor. At 0°F (-18°C), it contains six-tenths of a gram (only about a fifth as much as at freezing), and at -40° (on either scale), it contains only 0.07 grams (only a fortieth as much as at freezing). This helps explain why the heaviest snowfalls almost always occur when the temperature is not far from freezing, about 24° to 32°F (-4° to 0°C).
Any volume of the atmosphere will contain some water vapor, but only under certain conditions can it turn into snow. One necessary condition is that the relative humidity must rise to at least 100%. At this point (the dew point), we would normally expect the water vapor to start to condense (become liquid water) if the dew point is above freezing or deposit (become solid ice) if the dew point is below freezing. However, this doesn't always happen, and the relative humidity can actually exceed 100%, a condition called "supersaturation." Water vapor can usually change phase only if there is some object for it to condense or deposit on. At cloud level, there are such objects, tiny particles called nuclei, that are part of the atmospheric aerosol. The aerosol is composed of solid or liquid particles (other than water) that are so small that they remain suspended in the atmosphere for a very long time. There are many sources of aerosol particles, such as sea salt, clay particles kicked up by dust storms, volcanic emissions, man-made pollutants, and even the remnants of meteors. For water vapor to condense to form a liquid, one type of nucleus is usually required, called a condensation nucleus (generally water-soluble). For vapor to deposit to form solid ice, a deposition nucleus (generally water-insoluble) is usually required. Not all aerosol particles act as either type of nucleus, and not all nuclei are active at all temperatures. As a rule, the colder the temperature is, the more nuclei become activated. Important nuclei are tiny, typically only about a thousandth of a millimeter across.
Once condensation has occurred, the water droplets can continue in the liquid state even if the temperature falls below the usual freezing point. Such liquid droplets are said to be "supercooled." A third sort of particle is usually required to initiate freezing, called an ice nucleus or freezing nucleus (but some particles can act as both condensation and freezing nuclei). Again, the colder the temperature, the greater the number of substances that can act as nuclei. Testosterone, of all things, has been determined to becomes an active ice nucleus at 28°F (-2°C). (I can only hope it wasn't my tax dollars that paid for this line of research.) Presumably it is only at monster truck rallies and in NFL locker rooms that the amount of testosterone in the air is of meteorological significance. Maybe not testosterone, but other organic substances like pollen and bacteria can actually be important ice nuclei in nature.
Depending on the availability of active ice nuclei, supercooled water droplets and tiny solid ice crystals (typically less than a tenth of a millimeter) can coexist in the same cloud. It is in clouds like this that large snow crystals are formed. At temperatures below freezing, water vapor has a greater affinity for solid ice than for liquid water, so the vapor tends to evaporate from the water droplets and deposit on the ice crystals. This is known as the Bergeron process after Tor Bergeron, the Norwegian meteorologist who first proposed it. When the growing ice crystals become heavy enough to overcome the updrafts found in the cloud and fall toward the ground, they are called snow crystals. In layman's language "snowflake" is often used to mean the little six-sided symmetrical crystal, but to meteorologists that is properly called a "snow crystal". A snowflake is an aggregate of from two to several hundred snow crystals. At temperatures anywhere near freezing, snow crystals have a strong tendency to stick together (aggregate) if they touch as they fall, so snowfalls near freezing will feature mostly snowflakes. As the temperature falls, aggregation is less likely to occur. At temperatures below about 23°F (-5°C), individual snow crystals sometimes fall. Aggregated flakes rarely fall when the temperature is below 0°F (-18°C) and never below -33°F (-36°C). So you might be justified in saying it can be too cold for snowflakes to fall, but not justified to say it's to cold for snow to fall. Snowfall composed of individual snow crystals is sometimes called "diamond dust" if it sparkles in the sunlight, or "flour snow" if it does not.
At temperatures around -40° (on either scale), water droplets generally freeze even without active ice nuclei by a process called homogeneous nucleation. Without the presence of rapidly evaporating liquid droplets, crystals will grow by deposition very slowly and will have trouble growing to a size large enough to overcome cloud updrafts and fall to the ground. However, some crystals will by chance sublimate, and the vapor produced can be deposited on other crystals until a few of them are big enough to fall. If the updrafts are very weak, then very small crystals, often called snow grains, can fall despite their small size. In Japan, crystals as small as 0.07 mm (not much bigger than fog droplets) have fallen. Surprisingly, in very cold climates, particularly Antarctica, snow does not always fall from visible clouds, but from an apparently clear sky. This may involve a process called self-nucleation, in which vapor molecules come together by chance without benefit of a nucleus. This sort of snowfall is extremely light, but it can continue for days on end. I have seen anecdotal reports of this sort of snow often falling at or below -58°F (-50°C) in Antarctica, but I have been unable to find official records to confirm this.