So can you side with both the minutemen and the institute?

I know you can do railroad/minutemen, and I think its possible to do BoS/minutemen, so I don't see why you wouldn't be able to.

It seems to me that it would be the best option for the commonwealth, since the institute wants to preserve mankind in the long run but doesn't care about the day to day hassle, whereas the minutemen simply want to help and protect people in their daily lives but have no real plans for the future.
Not to mention you become the leader of both organizations, so coexisting doesn't seem out of the question.

From what I hear, you have to pass a Charisma check to remain on the Minutemen's good side after siding with the Institute, after that, you're all good.

Not the path I'd go, personally, but each to their own.

Ok cool, yeah figure most people are siding with the railroad or BoS but honestly don't really like any of the factions, the institute just seems the least cr*ppy.

New institute motto: "Good intentions, really sh!tty people skills"

It really depends on which community you ask. In places that I see with a larger sample size, it seems, with very rough estimation, that most people are going for or went for the Institute. This is mostly because the position that you are offered in the Institute, and people "roleplaying" that they can make a difference in this position.

Corrected.