That's the thing Alderan, you don't ever, ever have to drop $2000 for a very good gaming computer, far from it. Don't let anyone tell you or try to feed you that bollocks. In that range, you are paying premium price for new technology rather than performance/price ratio.
You can get a crazy ass powerful machine for under $1000 (with some serious processing power that will play all today's game on high settings). In the $700-800 range, you can build a very powerful gaming machine that also has some future proof (cross-fire or SLI as well as cpu upgrade). You can build a very good machine in the $600 range (a very respectable gaming computer).
When it comes to budgeting, a lot of this comes down to what resolution you plan to play at, if you're playing at 1680x1050 and less (22" and smaller wide screen monitors), you don't need to shell out for those cards like ATI's 5850s or 5870s or nVidia's GTX 470 or 480's, instead cards like ATI's 5770s or nVidia's GTX 460 will more than suffice. If you plan on playing at like 2560×1600 resolution, then yes you'll probably want to shell out the money on those higher end cards cards (of course if you're on a budget, this resolution is out of the question, haha). For "HD" resolutions 1920×1200 (typical 24" wide-screen monitors) you're looking at cards like nVidia's GIGABYTE GeForce GTX 460 1GB. This is of course for the newer cards, if you want to go crossfire or SLI, it opens up doors for many more options (and often, cheaper alternatives - though you do lose out on things like DirectX 11).
If you have the time or just curious, check out this site (I reference it a lot, a great baseline to start off while you do more research and reading - it's a great site):
http://www.hardware-revolution.com/700-gaming-build/