Well it has been soooo long since I've played the original NES game, and the first game for my GBA was Super Mario Advance, but it has now come to my attention that Super Mario Bros. 2 was just a dream. 
 
Read up this nice little article on GamesRadar about SMB2. It kind of makes me think of what crazy ideas the folks at Nintendo had back then for stories. 
Its a similar feeling that I have right now with Super Mario Galaxy 2. Its like "Mario went on this outer space adventure before, except it wasn't THIS Mario. It was a Mario from another dimension/universe."  Especially with that ending to Super Mario Galaxy.
 
Also, Super Mario Bros. 3 was a stage play?? How did I not see this before... 
 
And lets not even get started with Super Mario Land... Those are clearly the adventure of Mario in the 4th dimension.

 Ha! I never realised that about SMB3, but now you mention it, with the curtain at the very beginning, damn...
 
As for SMB2, I think I had some weird warped nostalgic memory of that game, I downloaded it to my wii and was madly disappointed, not that I thought it was awful, just that the love wasn't there any more =[ Nice article btw, Birdo is a dude? lul wut!