Is the weather system broken in your area?

Yeah, we get that a lot as well in my area. I believe most common weather in game (regardless of weather outside) is Cloudy, it was Cloudy on most days during winter and it's Cloudy even as I'm writing this message. But that has to do how the weather system in Pokemon GO works, this is a direct quote from here:
https://pokemongo.gamepress.gg/weather-boosts

"Pokemon GO appears to use forecasts by AccuWeather, which reports weather data obtained from various services, such as the National Weather Service in the United States. Pokemon GO assigns weather to geographic areas (“cells”) based on the hourly weather forecast provided by AccuWeather (emphasis on forecast: GO does not use the current weather). The active weather in-game is updated every hour, on the hour.

Due to the size of individual map cells and the innate inaccuracy of weather forecasts, don’t be surprised if the game’s active weather does not reflect real-world weather!"

These sentences explain why you are experiencing those kind of "issues" with weather in your area. For example that scheduled rain lasts for an hour just because the weather in game is updated once every hour.

The weather system could use improving that's for sure, but I'm still quite satisfied to it as for now. There are better things they should fix/improve first, right :)

Were I live it was sunny or at least partly clowdy all week. It never rained. Niantic saw it different though. They did let it rain a lot. Sometimes cloudy and when it was clearly sunny they gave us partly clouwdy...

In short we got robbed of weather boosted alolan eggys. At least with rain the fossils were boosted as well.

I've seen a weird weather: Symbol for partly cloudy, weather effects for rain (including for capture screen) but the boost was correct for partly cloudy. It would be funny if it suddenly snowed in-game mid July though (in countries where it's summer).

In Singapore, the weather can change from scorching sun to thunderstorm and back in 15 minutes.

I've found that it seems to get affected by the forecast rather than the weather itself. In a way it's been sort of helpful in real life. If it's clear out and the game shows rain, I take my umbrella with me to work since there's a good chance it will rain later.

Maybe there are relatively few points where the weather is determined in the game. The nearest one to you might be a few miles away or dozens. If it's farther away, that may be why it sometimes reflects the forecast (it might be raining at the point that determines your weather but that storm is still an hour away from where you are).