Often, you will see snow coming down when temperatures are above freezing. Or even though it was warm recently, snow sticks.
When snow falls at a heavier rate (comes down heavier), it has the ability to accumulate faster and override above-freezing temperatures.
Snow falling through above-freezing temperatures begins to melt. Because the flakes absorb some of the surrounding heat, it causes a cooling effect for the air around it.
With the air now cooled, snow after that can fall as snow all the way to the ground, and over time, stick.
Heavier snow will allow for this effect to occur. However, with a light snow, this process is very slow to occur. Many times light snow is at the end of a storm, or in some other situation where the snow is brief. Because of that, accumulations are much harder to come by, especially if temperatures are only at 32, and especially if they are 33 or 34.