Keep in mind that the interior of the continent of Antarctica is one of the driest deserts in the world. That's because at extremely low temperatures (near -40C which is equal to -40F) the ability of the air to retain sufficient water vapor to precipitate into snow is to low. But it's only at these low temps that your phrase is partially true. Other conditions that exist at warmer temps can also prohibit precipitation, but not neccessarily just cool air
It also is somewhat misleading, "too cold to snow" often means in that it is High pressure and bright and thereforethe conditions for snow are unlikely to be met Ie clouds etc.
In Britain the coldest temperatures tend to occur when an area of high pressure settles over the country. This usually means cloudless and windless conditions and very cold nights. Lack of clouds of course means lack of snow.
Snow occurs when a low pressure system brings in plenty of cloud. The cloud keeps the temperatures up a little but precipitates as snow, usually when the air temperature is a degree or two either side of freezing.