I recently heard a local weather forecaster (was it Matt Noyes on NECN?) address that question. He said that it would have to be colder than it ever gets around here (-50F?) for that statement to be true.
He went on to give an explanation of how sometimes moderately cold temps can create conditions which is not condusive to snowing, but I forget the logic he used.
basically its a function of how much moisture the air can hold. The colder it is, usually the drier it is. So those oldtimers really should have said, "Ayup, too dry to snow". But that's just not as catchy.
I wake up in the morning and its forty below
the weatherman tells me it's too cold to snow.
I look out the window and I know that he's lying
My car's out of sight and the snow's still flying...
I always thought it was a reference to no clouds to hold the heat in, hence on those bitter -5 and colder days you would not get snow unless a cloud cover came in and warmed the area up a bit by retaining/reflecting some earth heat (and brought snow...?)
But now that it was mentioned, I recall cold air being very dry versus warm moist air, example (that was explained to me once) being the warmer moisture-ladened winds over Lake Ontario hit the foothills of the 'Daks by Watertown and as they blow over the tops, getting colder as they go up, they dump all that moisture in the form of snow.
I hear this often to mean that extremely low temps, -20F to -40F (the latter of which I have not seen in years), are accompanied by very low humidity, hence no snow.
The optimal temperature for snow growth, fluffiest flakes is a temperature of 15 degrees F. So once you get lower temperature they get smaller and have more ice. Above 15 they get wetter. Once you get to temps around -50 it turns into only ice crystles coming out of the sky. -Mattl