Could it be, perhaps, that the BoS are working *with* the Institute? To me it depends on one thing...
Are the BoS part of the Lyons division, or not? So far all evidence says yes, though airships is the one thing that throws a chink in the chain, being they may actually be midwest.
Even if it is the Lyons division or not, Lyons is focused on helping *people*. Who knows if they care of Androids, but my bets on they aren't. This could mean that if there really are more members of the Preston Garveys Militia that they may side with the BoS seeing as what early (and small, mind you) footage we have seen shows them helping humans.
What better way to help people, then with armed and dangerous synthetics who, in my opinion, don't have freewill unless programmed with such freewill? So many questions... Its really hard to say until we get the game in our consoles/PC's.
My one wish? That Liberty Prime is back in action. As my signature may give a clue to, he is probably the coolest things I have ever seen in Fallout, and his anti-communism ramblings are amazing.