I think you can and the reason for that is that I used to think I couldn't but I did, it worked and I fix them for a living now so I've probably built a hundred or more as well as repairing.
Keep and open mind and have a look at http://www.novatech.co.uk. From there you can buy a barebones system, so you have only to add components on to what they ship you ready assembled. That way you get to specify the graphics, for example, and just plug the card in when the box arrives.
You'll get a great feeling out of making your own computer and your £3 to 400 budget will go a lot further.