Oh.

It is such a widespread weather lore through many cultures that it has some credibility.

Sunny weather implies High Pressure - and at this time of year that usually means cold.
It's likely that before any warmer conditions arrive there will be at least one or two stormy spells at a transition to warmer air masses.

If it is dull and wet/snowy we're already in an unsettled changable period and warmer air could move in without it getting worse!

What doesn't make much sense is tieing it to a particular day - for example tomorrow ill be fine for many here but it isn't due to a stationary High but a very transient affair.

The period around mid-Feb is often a key turning point as the extreme cold usually finds it more difficult to extend than is possible in December and January with the minimal solar energy.
The sun starts gaining strenght rapidly through February.

If February is wet, then Spring will be Wet.

If it thunders in February, It will be frost in April.