No, Only Swift Swim + Drizzle on the same team is banned, Sand Stream + Sand Rush/Drought + Chlorophyll on the same team are perfectly legal in OU. Seeing as all of the weather starters are in OU, this does not apply to lower tiers.
Banlists
OU
•Banned Pokemon - A team cannot have any of the Pokemon found in the following tiers: Ubers.
•Drizzle + Swift Swim - Using both a Pokemon with Drizzle and a Pokemon with Swift Swim on the same team is banned. - Smogon
No where does it say that those cominations are banned.