Is it possible to bootstrap at least an approximate 6 month LIBOR curve (actually NIBOR, for Norway, in my case) if rates for 3 month FRAs are known? For example, say we know the rates for 1x4, 7x10, and 10x13 FRAs. Can we deduce any points on the 6 month curve?
No, you can't. You can never deduce the 3M/6M basis spread from 3 month instruments alone.
If you consider the OIS curve riskless, you can interpret the 3 month curve as riskless rate + additional cost for things like credit risk, liquidity and so on. The 6 month rate contains even more of these credit risk and liquidity cost. How much exactly though is impossible to say from 3 month rates alone.
You would need spot rates to anchor on. For example, in addition to the forward rates, if you have one-month spot, then you can calculate four-month spot based on your forward. you can impute the spot between one and four months.
To your question specifically, if you have spot at one-month and seven-month, then you can impute the six-month spot. Other combination like four and seven would work, too. The idea to is have spots on both sides so that you can impute.