Nate Silver (of 538) has some space that he's dedicated to this effort in The Signal and the Noise. Randal Olson's reproduced some of that related to current-day abilities, which show that we're currently able to give better-than-random results for a few days in advance, but not much better after that. And, unsurprisingly, data beats expertise when it comes to accuracy.

A good deal of the data-collecting tools have been developed or implemented relatively recently, and that seems to correlate with improvements to short-term forecasting, to the point where a five-day forecast in 1991 was roughly as likely to be accurate at a three-day forecast in 1981.

They've improved enough that they can probably be trusted to determine whether you should bring an umbrella tomorrow, but the historical numbers and especially expertise-based numbers were inaccurate enough to explain the origin of the meme.