Yes and no.
The answer to your question has to do with stimulus and adaptation. Training applies the stimulus, and the body adapts to be able to better handle that stimulus. The higher the reps and lower the weight, the more the stimulus trends towards muscular endurance adaptations. If you do not have enough muscle force output to handle the weight, then the body will adapt by increasing muscle size. Once ample force output is achieved, then it will continue to adapt in ways that allow you greater endurance, of which, hypertrophy is not necessarily one.
Consider an endurance runner, who does thousands of leg extensions with partial body weight. At first his legs will grow to accomodate the weight, but do you see endurance runners with massive quads?
So, in brief, lifting heavy stimulates the need for greater force output to handle the load the next time. Muscular force output is directly related to the cross sectional area (muscle size) of the contractile proteins. Continue to challenge the muscles with a progressively greater requirement for force, and they will adapt by increasing in muscle size (progressive overload).
Of course...that is a very simple explanation, and if I don't make mention, some other educated muscle head like Rodja or Josh will mention something about increasing force without an increase in muscle size via neural adaptations....but that's another story for another day.