Warning: Spoilers ahead!

Like many anime series, Cowboy Bebop has an underlying thread, a message that permeated each and every episode, and foreshadowed the conclusion long before the last episodes. IMHO, this thread is that You can't run from your own past, for it will always come back to haunt you. In order to get on with your life, you have to take deal with your past, and take responsibility for your actions...

Each and every episode revolves around this one thread. In the first episode Asimov uses the drug bloody-eye to get into a state of high metabolic speed, freezing time around him and allowing him to temporarily escape his pursuers. But when Maria is hit with a bullet and her "pregnancy" is revealed as a blind to hide the drug, it is apparent that their intended bright future never had anything new or bright about it: one cannot create a future by running away, the past one is trying to escape will eventually abort one's hopes, which never honestly existed anyway.

This is true even for the comedic episodes, such as Toys in the Attic: Spike's laziness to clean out the fridge comes back to haunt the crew.

Consider the fates of the main characters: Faye learns about the past she forgot and escaped from, but desperately longed for, and sees that everybody she might have loved (family, friends) are dead and her house is destroyed; Jet loses his best friend Spike and learns that his old best friend was a traitor; Ed finds her irresponsible, and quite possibly insane father (note his eyes), trying to ensure the future by mapping the ever-changing surface of the Earth (while properly caring for Ed would probably be the better investment into the future); and finally the cornerstone, Spike who ran away from the final confrontation with Viscious by pretending to die, only to be caught up in the past he was trying to evade, losing Julia and ultimately dying while fighting Viscious. The point is that when you carry a heavy weight, that weight eventually becomes a part of you that you can't leave behind or move past. The past, and all that comes with it, cannot be unwritten. And so, in the end, the fellowship is broken because of errors made in the past.

Still not convinced that CB is really about taking responsibility for one's past? Let the last CB title card speak for itself:

You're gonna carry that weight.