$1,500 is more than enough IMO unless you are spending $500 on a graphics card.
My computer was built for $900 and is great and quiet.
A few obvious suggestions IMO:
Grab an SSD. This is where the bulk of my money would go. I never understood how big of a difference it would make until I bought one.
Corsair makes easy to install water-coolers that are ultra silent and work really well.
Get RAM that works with your motherboard. Motherboard manufacturers list the exact model #'s of RAM that they tested on each specific mobo. Get RAM that is on that list.
Get a quiet PSU.
Get a case that has flexibility. I have
the Corsair Carbide 500R (in black). I had to replace one of the case fans as its noise was annoying, but other than that, had plenty of room, came with everything I needed, has an easy mount for hard drives, and tons of cable management built in. Also has a fan speed switch so you can turn it up when doing something intensive, turn them down when you want the PC to be quiet. Corsair also provided an area at the top specifically for their H100 and other dual-sized water coolers which is nice.
Did I mention how awesome having a silent computer is?
Anyways, just be sure to do your research on the parts. Obviously, just about every product has at least one good and one negative review so its hard to judge. Read up on the negative
reviews and try to research them if you can (doesn't work with _____, too loud, etc). Make sure that you have RAM that isn't faster than what Intel supports. If you use a 1600 MHz RAM on an Intel IVY Bridge CPU, they won't cover any damages that happen. Same with overclocking, etc.