Purely time-wise, I believe I could have four more lovers. It's for the emotional part that I'm not sure how that would work (I'd need to fall in love with them first, after all).
Then again, I'm unemployed. Add a job, and I'd have much less time. And my main problem I think wouldn't be being with them one on one, but with all of them at once. I'd want everyone to know everyone, and if possible be in good terms, and I'd want to do things all together... And I tell myself, I'm not a very social person, wouldn't that feel like too many people?
On the other hand, we were 6 in my family (4 kids + parents) and we did lots of stuff all together, and it was fine... And that's just one more person...
Six would be much easier to achieve if it included my metamours. Then even if both my partners only have one person over me, that's already four people. If I get a third and they have a partner, or I get a third and Raga or Sean gets a third too, or if they both get a third, we're at six people over me. And I'd only be involved with two or three myself on a romantic level.
I think at that point, the way to go would be doing a lot of things as a group, or with several people at once. Still trying to get alone time with everyone, in which case you split into couples (with one trio I guess) to do various activities, then get back together and shuffle, then have supper all together, or something.
It seems to me in your case you're all a big family, so it's not 6 unrelated people with you in the middle, which makes it all easier in this specific case (more complicated in many ways, too, I'm sure!)
And I'd also be sure to get some alone time. But I think I probably wouldn't have much of a social "external" life, one of my lover would almost always been involved in some way! That's not necessarily a bad thing though.