Teams like that are based on very, very fragile foundations. Scales can tip very fast the second even one of your weather Pokemon falls.
For example, you're out on a battle where your surviving members are Venusaur and the sand duo, and Venusaur needs to heal using Synthesis. The second Sand Stream starts on your field, Synthesis only gives out 1/4th of your health and throw in residual damage, which will end your leafy dino.
The basic point is that, for every weather you will have only two benefactors, the rest will all suffer from anything but their weather. It makes the game much harder to win, and often you have to just sacrifice your weather setter simply to set up the weather you need, which may or may not let you win.
Another problem is that switching in doubles is scarce. It's fast paced enough to have you think twice before switching out to a different weather setter, while the opponent could just fight out the weather and single out your Pokemon. It's often a better idea to stay in and save the Pokemon you have through Protect and all than making haphazard switches.
The winning point of normal weather teams in doubles is that, because of the fast pace, the weather's 5 turns can make the whole difference. In a multi-weather team though, this can backfire, and you could end up with your Kabutops tanning in the sun, and Venusaur flailing in a sandstorm.
Lastly, weather-casters are put under a lot of strain. Not only will they're going make a difference, but they will be the first targets in the opponent's mind. They will take down whichever caster they think will hinder, and leave the one that will start a weather they would enjoy more. A Mamoswine would love to eat your Jolteon/Ninetales with an Earthquake and much rather go in the sand.
TL;DR: Multi-weather teams are too fragile, it's almost always better to go with single weather teams.
Hope I helped!