Why is Euclidean distance not a good metric in high dimensions?

I read that 'Euclidean distance is not a good distance in high dimensions'. I guess this statement has something to do with the curse of dimensionality, but what exactly? Besides, what is 'high dimensions'? I have been applying hierarchical clustering using Euclidean distance with 100 features. Up to how many features is it 'safe' to use this metric?

[yoshihiroueda]

個々の要素間の距離が互いに離れていき、近い点と離れた点の比が1に近づく。