Before this year, when we started home schooling, we did one sport per season + gymnastics year round. So....swimming in the summer, soccer in the fall, just gymnastics in the winter, baseball in the spring.
Now that we're homeschooling we do swimming & gymnastics & aikido year round...plus the "seasonal" sports, and we'll probably toss in sailing this summer. Plus, of course, his outside classes (3). He also skateboards, bikes, etc.
Now, we have only one child (I see you have five), and we homeschool (have I mentioned that about 10,000 times already yet?)...so his sports are times for him to just be a kid with other kids, and to have time to interact with those kids and adults without mum & dad around. Super important. In any event, here's my yardstick:
1) Does he have a huge grin on his face the whole time?
2) Do I actually WANT to take him there?
If EITHER of those are answered "no" we phase it out. I'm in school, too, and there are quarters where I can do my HW in the car, and quarters where I Just. Don't. Want. To. Leave. The. House. for anything short of a fire, because I've FINALLY gotten time to clean the darn kitchen, but it's time to get in the car for "x". If I find myself loathing "x", we either tweak the schedule or drop "x".
As a kid in a large family, there were years where we each got to choose ONE activity, and years where we each got to choose TWO. It was a question of $ and my mum's sanity. The deal, once we were old enough to complain, was that we could do as many things as we liked, if we paid for it and had transportation to and from. So in HS, we all took advantage of the school sports/activities (swim team, drama, etc.). Prior to that, in middle school, I became a working student at a racing barn in Del Mar, in large part because I could cut through the canyon preserve and walk it in under an hour. We also knew that if we were whining about something special that was given to us ( as in "but I wanna do thaaat toooooo", what we already had would be taken away out of ungratefulness.
In any event, in my humble opinion, I think what works...is what works for your family. We all have different needs and tolerance levels.
:) Good Luck!