2 - 3 has been a limit. The few times that I've gone beyond that limit I tend not like the results.
I tend to let a RP sit at the back of my mind as I'm going about my day. Planning, what could happen next, or what I'm going to do next. My headspace stuck in one of the RPs, only to get a response from RP 2 or RP 3 or RP 4 first. I then have to pull my head out of that Headspace to get into another. And then sometimes to have that replied come in as I am in the middle of a response.
I have always had a rule with Mutliple RPs with First Come first Server. And I don't like it when I am tempt to pause my response to the second, as my current headspace might have that response already mentally written, but sometimes that happens cause its easier.