Short answer: yes!
Long answer: it depends...
I think this runs closely parallel with the usual answers to "should I build a kit plane?" - ie if you think you'd enjoy actually learning about and building a PC, then do it, *but* if you're just hoping to save a few quid on a new PC, then don't.
I haven't bought a complete PC since about 1995 - that was a 90Mhz Pentium with a new whiz-bang technology called 'multi-media'
Since then, I've just upgraded, upgraded, upgraded - I think the only things that are still original in the whole set up are the keyboard, mouse and an IDE cable! I enjoy doing the upgrading and it's allowed me to only upgrade the bits I wanted, when I wanted, without having to worry about copying data/software/setup from one machine to another, but it hasn't saved me any money. On the other hand, I've built a second PC (on which I run Linux) using many of the 'old' parts from my main machine.
A good way to start is probably to get an old machine from somewhere and take it to bits. You need to get familiar with what all the parts look like, how they fit together, how much force you physically need to use when slotting the (eg) video card into place, etc etc. Once you've got that basic level of competence with all the components, it's actually a very quick and easy process. Provided you order all the correct and compatible bits and pieces, an experienced PC-builder would be able to put them all together and have a working machine in two or three hours.
It's a lot easier these days now that most motherboards have 'soft' setups - when I was a lad, you had to spend a couple of hours just reading the motherboard manual and fiddling with tiny little bits of plastic called 'jumpers'...
cbl.
ps be warned though, this can be addictive - some people end up spending every hour of every day trying to wring the last ounce of performance out of their home-builts - they are known as 'overclockers'. They remind me of audiophiles who spend more time tweaking their stereos than they do actually listening to music...