Alone does not mean lonely

First, lets have some definitions about the words themselves.

Alone - having no one else present, on one´s own
Lonely - sad because one has no friends or company, without companions, solitary
(from http://oxforddictionaries.com/)

All of us felt both of this emotions - but why sometimes Alone stands for Lonely?
Is it inevitable? Is it us who let this feelings to come on our surface?
Or is it a pressure from society that people need to be in groups?
Can you be alone and happy at the same time? Forever?