>it doesn’t seem like anyone really trusts anyone anywhere anymore
No offense intended, but have you considered that maybe you're just surrounded by bad people? I have a reasonable amount of trust and faith in the other people that I work with. If I needed someone to pick me up in the middle of the night from the hospital, I'd trust my coworkers as much as I would trust my wife or friends.
My company is hardly some "golden child", but it generally does right by its staff, and the staff, more or less, does right by the company. Things aren't perfect, but I would say the amount of inter-office trust is about as close as anyone would reasonably expect. Problems pop up, but with a healthy amount of addressing things head on, those problems don't rot and fester.
Need someone to be honest with you about the deadline of an upcoming project? Done.
Need to get a hold of someone off-hours on a three day weekend? Alright, as long as it's something important and time sensitive.
Need realistic feedback about a presentation you're working on. Sure.
It all revolves around mutual respect, camaraderie, and clearly defined roles. When I see an upcoming issue with an area I have nothing to do with, I don't feel the need to fix it "because no one else will", but, I also don't leave it to blow up in someone else's face either. I let the right people know, or tell my boss if I'm not sure who should be involved.
If I had to take a guess, maybe you work for one of those larger corporate type companies with lots of middle managers, office politics, etc.? I'd encourage you to look at getting away from that if it's weighing heavily on your ability to trust others. It's something that I dealt with for a length of time, and once I moved onto a smaller shop, the ability to trust and genuinely rely on others improved quite a bit.