Question: As they say, "politics is the exercise of power", and "power is the ability to make things happen in an organisation".
I am working on a project which has some overlap and is partially in competition with other projects. Managers at all levels have different ideas about what should happen. I am trying to understand who is supporting whom, what are the networks I should be aware of, to have a clear idea of where support for our idea might come from.
My line manager doesn't want me to think about politics at all. Other people told me not to be political. But I think these people might have their own agendas - like everybody else. I have seen what happens when you work hard on a project, ignore the politics, and then get backstabbed by some unforeseen person or project.

I do believe that "ignore politics" is bullshit advice: whether you manage people or build things, politics will determine the allocation of resources to your team or project, but I might be wrong.
So, let's ask it this way: **"what are the circumstances when ignoring company politics is good advice?"**

Answer: 
The specific answer
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Your line manager knows the situation better and wants you to stay back and stay safe, "ignore politics" is just a phase they use. As a non-manager your advices are appreciated but you shouldn't get tangled up because there are understandings among participants that you might not get. 

The generic answer
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"Ignoring politics" is a inherently a political decision. You want to go this route if you value your workplace stability over the value of your work. You wouldn't get into workplace politics at a summer job, you may not want to get into workplace politics shortly before retiring.  

You need to ignore office politics nevertheless when you would rather leave your job than go with one of the proposed solutions, especially if it is illegal and would be incriminating to you (e.g. "Should we outsource medical data processing to a company with a questionable reputation?" "Should we pay dividend before announcing bankruptcy?" or "Should we save on untreated waste disposal?").
But also: *what is* **your** *job?!"* 

"Politics" obviously surrounds "every single human activity." So – is this *actually* "a battle that you have to fight," in order to fulfill your business obligations to the company?

"If it isn't – *butt out!"* Do your job. Earn your salary. Dazzle your [internal?] customers first, and your managers second. Don't worry about the rest. *"It's above your pay grade ..."*