It is possible, as much as back in 2010. The problem is not so much what you can and what you can not do with a tool. I can do much of everything I do in illustrator using a text editor. I can do much of all pixel pushing in any software other than Photoshop again even an editor.
However, I need to be compatible with other designers and users. Its not about having the capability to do something it is about the capability to do something with others. And here Adobe has the design crowd hung up.
Lets look more in detail
Photoshop
So if you look for a application that can do pixel pushing, as in drawing. No contest Photoshop loses out on competition. But that is until you realize that Photoshop is not a drawing application, its a photo manipulator. But even here Photoshop is not hard to topple for many tools that work more like Lightroom is worth more to many.
So unless your looking to publish stuff in print you will find that Photoshop is by far the easiest one of the lot to replace. It is the integration between printing vectors and pixels where Photoshop comes out as king. Not because its a spectacular tool, no but because of its breadth as a tool and its printing/color management capabilities.
Unfortunately you still need to be able to open many a Photoshop document as this is what the data was authored in.
Vector drawing
Well lets just say that Inkscape has a crappy user interface. For Illustrator there is really no contest. It is also the application i would like to replace the most since adobe can not seem to get anything right in their illustrator development end.
I man can inkscape do what illustrator does? To a certain extent, illustrator has better PDF support and EPS support, Panatone support. In many ways i can replace illustrator with notepad more easily than with inkscape. Again its a question of compatibility. Illustrator is a great tool when you want to debug stuff that is somehow broken. In ways that inkscape can not match. On the other hand inkscape has better support for SVG. But not much.
So what do do I replace my illustrator with? I replace it with an older version of Illustrator. I have lots of choice when it comes to replace photoshop but not a single good contender that would work on my setup.
Page layout
Well InDesign is hard to beat as is TeX. But personally i use InDesign far too little to be an authority on the subject. Indesign has pretty nifty functions that only high volume industry can appreciate like the inbuilt XML publishing system. TeX on the other hand is hard to beat if you want to do math typesetting. But these hardly compete in the same arena. Don't know much about Scribus other than using it was a exercise in pain.
Epilogue
All in all the adobe suite is pretty cheap. And while i am looking for an alternative there is really nothing that has even fraction of the features I use. But i am a special case hopefully you consider yoruself a special case too.