No.... Auths will only get banned if someone else uses them on a different IP. Even so, you can ask for another issue, so really, you never lose your auths.
As long as all your auths are running from the same IP address they will not get disabled.
This means that with certain internet providers you might have the same IP's on different computers in your house so you could still run the auths without getting disabled
rsbots.net says 1 auth per account, however if you use them on the same IP they have know way of knowing, so you could run 10 SG bots on one auth and they will never know so long as it's on the same IP.
I can't recall rsbotting, but I don't think they have the 1 auth per account rule, you can use it for as many accounts as you want.
Either way it doesn't matter, keep the SG bots on the same IP and it's all good.
Auths are managed by Impsoft (Company that owns Nexus client), NOT RSBots.net/individual websites.
Where do you enter the auth code? Nexus client. Does RSBots.net have access to the client/client information? Highly doubtful.
The whole "1 auth/1 account" rule is ridiculous bullshit on their part. It's just a ploy for the greedy bastards to get even more money. It's not bad enough they're reselling other people's scripts for more expensive, but they spread this lie around, that many people sadly believe.
tl;dr Nope! As long as all the bots are being run on one IP address, you can have 600 different bots running the same auth at the same time.