k so here's how bans work: they aren't actually applied until the next time you log into the account. So for example:
>I bot on an account for 75 hrs straight
>I get reported some time during that, jagex places a delayed ban on my account to ban me in 2 days
>I quit the game or quit that account 1 day after i stop botting (1 day before my delayed ban is implemented)
>I do not log into the account for months - My account is still technically NOT banned, as in the timer has not started yet.
>As soon as i log into my account it will be force logged out and the ban will be implemented, whether this is 2 days after or 2 months after.
They do this because they don't want people just brushing over their 2 day bans if they're busy over the weekend or whatever. They want you to not be able to play on the account when you want to play on it. Also, if they made it so your ban timer starts ticking down while you haven't logged in, then some people wouldn't even know that they got banned if they didn't log in for a few days.
So pretty much that's probably what people are misinterpreting as "getting banned a month after botting" as.