Why do boiling points of alkanes and alcohols become closer with increased chain length I came across this graph, which clearly shows a large difference between the boiling points of alkanes and their equivalent alcohols. 

However, it is clear that as chain length increases, the difference in boiling point between alkanes and alcohols decreases.

What is the main reason for this? 

I have read that it is due to the dispersion forces increasing in both alcohols and alkanes, but wouldn't this not really have an effect on the hydrogen bonding of the alcohol? (meaning that boiling point should be consistently higher)
It is simple. Longer the chains higher the similarity between them. In other words the OH group are getting "diluted".
It is more difficult for the donor to find the accepting counterpart, and the number density of the hydrogen bonds, both for volume and mass, gets lower.
The interaction between molecules becomes dominated by VdW forces. 

To visualize this think of methanol and of a very very long primary alcohol. In the second case it is somehow "more difficult" to spot the difference versus the parent alkane.