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: 
"Ignore company politics" can have two meanings depending on context. Walking into situations oblivious of the politics is a terrible idea. You are creating a situation where you are way too likely to shoot yourself in the foot. 

When someone says "ignore company politics" it can also mean don't put active effort into leveraging politics. Knowing the lay of the land and actively manipulating the situation to take advantage of it are two different things. Ignoring politics is good advice in organizations where the processes and culture are reasonably efficient and there isn't much competition for resources. If an idea will stand on its own merits and doesn't need a powerful ally to proceed then putting effort into gathering allies to support your idea is a waste of time. In the organization I'm in right now this is mostly the case. I've seen pet projects shut down hard and I've seen projects that nobody was excited about make it through.

The big part of this advice that you may be ignoring is if you are seen actively pursuing goals which appear political you impact your image and your credibility. You really don't want people to question your hidden agenda when you need something from them. This is especially large risk for people who openly talk about workplace politics.
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?").