Does 很 actually convey a meaning of very/extremely? In dictionaries 很 is translated as 'very' but I'm not sure this is correct.

If I want to say "I am happy" the mandarin is 我很高兴. The character 很 is used even though the english phrase shows no sign that I am very happy or extremely happy.

I have been told to say 我非常高兴 if I am feeling more happy than normal

So does the character 很 show any indication that I am 'very' happy?
很 is often pronounced lightly, so it is actually without any real information. It seems to fill the role of a predicate marker (e.g. 'copula') with adjective phrases. However, if it's stressed, then it does mean 'very'. 
(Mandarin Chinese does have stress; however, it seems to get ignored in many language classes.)