To add to this, there are large-scale structures that DO make excellent predictors of weather to come. For instance, a "bomb cyclone" just hit the western coast of California. These mid-latitude cyclones follow somewhat predictable patterns.

Over the US, winds tend to blow from the west, especially along the Pacific coast, so it was pretty much guaranteed that this weather system would move from west to east. In that link you will see a satellite picture of the storm, which has the tell-tale cyclone shape, like a comma with two tails. The clouds on the western tail form a narrow band, which is indicative of the tall storm clouds associated with a cold front. The eastern tail is wide, made of low-hanging stratus clouds that form due to an incoming warm front. In the center of the comma is a swirling occluded front where the cold and warm fronts meet.

Experiencing a few days of little precipitation and overcast skies does not necessarily tell you a whole lot about the weather to come. But if you know that the large scale formation looks like this cyclone and you experience those overcast days followed by a couple clear days, then on the clear days you can bet that you are in the cloudless space between the comma tails, so the cold front is coming, bringing with it heavy rain.

But as HankScorpio here said, using monthly averages as a prediction tool is not a great idea, simply because of the potential variability of year-to-year climates.