Jim Roseberry
Lining the case (or at least the lid) with sound deadening foam mat can cost a bit but can be worth it. Plenty of DIY case quieting info available on sites like silentpcreview and htpc sites if you search for it.
Lining the case with foam can help vibration related noise, but it won't make loud components quiet (as I'm sure you already know).
No different than putting that 1/4" foam on your studio walls... and expecting it to sound-proof the room. 
Jim, there are different kinds of noise-reduction foams, performing different functions.
Lining case panels with noise "dampening"or "insulating" material (such as a closed cell foam like neoprene sponge) can reduce panel vibration as well as noise transmission through the panel. A layer of vinyl flooring (or another "limp mass" material) can also work in this role. Such noise-dampening/insulating treatment may be found lurking behind the carpeting at the front foot wells of automobiles (to block noise coming through the thin sheet metal chassis into the passenger cabin) and inside old high speed printer enclosures (to block noise transmission from inside).
Using a noise "absorbent" foam liner (typically an open-cell foam) inside a case can quieten the case by absorbing the sound (converting the sound waves' mechanical energy into heat) and diffusing the sound (reducing reverberation (reflection) inside the case).
Generally, the idea is to dampen vibration of large panels (e.g. the side covers) of a case, along with using sound absorption treatment to reduce noise inside the case (which if untreated can be emitted through vent openings).
Baffling of vent openings can also help to reduce sound emission from a case, as can designing the case to avoid direct paths to the exterior in the first place (e.g., baffled front panels with side or bottom intake openings, ideally being lined with noise absorption material) as well as using quieter fans on heatsinks and especially at intake and exhaust vents.
The noise spectra blocked/absorbed and effectiveness (loss coefficient) varies by foam material and thickness, of course, with lower frequencies being more difficult to treat for. Thicker is generally better, but using a thick lining can be problematic as it can make case panels difficult to mount, make cabling difficult and may also reduce cooling air flow inside the case.
There are "composite" foam sheets available combining both a noise dampener layer and a noise absorbent layer, or one can DIY and laminate different layers.
The purpose and type of sound treatment used in a studio varies, being for noise diffusion or noise absorption or blocking noise transmission, or combinations thereof. But no amount or type of sound treatment inside a studio will help if your studio DAW PC is noisy (except for an iso booth)!
There, more than you ever wanted to know!