simple fact: most cases designed to be silent are not cheap due to the thicker construction and/or added features to reduce noise. most cheap cases will be a bit noisy considering the thinner metal they are made out of.
generally you achieve case silence through a few methods:
-thicker steel or aluminum construction than normal
-sound deadening material on side panels
-large 140mm silent fans with rubber dampers
-mounting hdds with rubber dampers
-rubber case feet dampers
-no window with soundmatting on the side panels, minimal venting mesh.. just fan holes.
-flat front cover with foam for a fan filter and noise reduction.
and hardware matters too:
-use of double or triple fan gpu heatsinks over blower design. the asus dcuii does good but the tri fan designs are better however the gigabyte reliability is questionable and sapphire is only for amd cards and normally a little pricey. you do have pny and a few others offering tri-fan designs but they are generally not sought as much as the major brands.
-use of quiet cpu cooler. larger than necessary cooler means lower fan rpms and lower noise.
-use of ssd drives not hard drives. ssd are silent.
-if using hdds using normal variants not something like 10k raptors.
if you did get a cheap case you could try dynamatting it (or using the cheaper secondskin, fatmat, and b-quiet brands) to reduce noise however they only work so-so and much depends on how the case is built and what hardware you have, etc.
if you want a silent pc case... one that impressed me was the nonoxia deep silence
http://www.amazon.com/Nanoxia-Silence-Computer-Motherboard-Controllers/dp/B008J1I0I6 which had some really heavy soundmatting inside.
the deep silence 2 beats out the fractal r4 for both cooling and noise tests by a bit.