Here's a good RP reason: you fully believe in preserving and studying technology, but you also happen to think that synths and non-feral ghouls can be people, too.
Also, yes, you can. I was part of the Minutemen, the Railroad (and by extension the Institute as an undercover agent), and the BoS at the same time. The only way it doesn't end in almost everyone dying is if you ultimately side with the Minutemen, though. But in my main playthrough, having completed the game, the Prydwen is still in the sky, I'm still a Paladin of the Brotherhood, and the Railroad is still running synth survivors of the Institute out of the Commonwealth. In fact, I just helped train a new squire and then headed over to destroy an anti-synth gang for the Railroad. While I was on my way there I stopped by to help defend a Minutemen checkpoint that was being overwhelmed by Gunners.