An interesting topic on which I would also like to share my experiences with (and basically add onto what Phil has stated).
I think we all know that different users have different needs in terms of what they wish to use a computer for.
Most users use a desktop computer to browse the internet (which includes Facebook, Skype, etc.), check and compose e-mails and perhaps for some office related work.
For that, you do not require the latest and greatest PC to perform those tasks.
To provide some examples of where I installed a "modern" operating system (being anything post Windows XP) on an old PC:
About two years ago, I built up a PC for my aunt who required it for "normal" daily use.
This PC was an old Athlon XP 2000+ on a VIA based (I think it is a KT333 chipset) motherboard.
Surprisingly, older VIA chipsets are still fairly well supported (in the sense that there are at least drivers available) under Windows Vista & Windows 7.
I installed 1 GB DDR RAM and an old X800 ATI graphics card in the PC.
At that point, I decided to install Windows Vista on it, but if I had to do it now, I would just install Windows 7.
After installing all the other software (anti-virus, Office 2007), it still runs quite well.
Recently, I acquired another old Pentium 4 PC (Socket 478) with a Pentium 4 2.4 GHz (with Hyper-Threading Technology).
I plan on installing this PC for my uncle, since he currently has an old PC that still runs on Windows XP and needs a more up to date OS (his hardware still has an Nforce 2 chipset, which doesn't have drivers post Windows XP).
The motherboard is MSI based with a VIA P4M800 chipset.
This PC isn't actually "newer" that my uncle currently has and the onboard graphics card doesn't actually have drivers post Windows XP, but by installing an "external" graphics card (I plan on installing a GeForce 6800), this problem can be overcome.
I also installed 1 GB RAM on it.
This time I went straight with Windows 7 and it runs fine for normal use.
So, depending on your needs, and if your hardware still has drivers for a "post XP" operating system, then there is probably no need yet to go and fork out money for a new PC.