Thick Walls
Thick walls definitely cost more to build than thin ones, but the benefits are also great. Walls have three uses: slow the transfer of heat, act as sound barriers and divide spaces. Thick walls give a building a sense of solidity and protection from the elements. They also increase comfort by raising the mean radiant temperature of the room, and create deep window sills that are great places to put plants, a coffee mug, the book you're reading etc, as well as providing a great sound barrier.
Thick walls also take up space, which on small urban lots can be a challenge1. In all but mild climates, exterior walls should be thick due to holding extra insulation: anywhere from 7 to 16" of it. By using foam board, thinner walls can be achieved, although there are environmental tradeoffs (see envelopes in the energy section). Thick walls also are challenging in that wider sills, jambs and liners are required for windows and doors, but enough builders have done thick walls that there are a variety of solutions available. For new construction, thicker foundations, or wider slab edge footings are often used (in spite of the fact that the second wall is supposedly non-bearing).2 For retrofits, the additional wall is usually hung off the existing one, either by adding a foam board skin, or using Larsen trusses.
Non-insulating thick walls: the concept of thick walls can be used internally as wall, both as a way to create storage and as sound insulation. Simply adding tall cabinets or a bookcase will create some sound insulation. If the area is thickened quite a bit, you can create alcoves within this area Closets also act as sound insulation and often bedrooms are divided by wall of closets for this reason.
If the goal is soundproofing, you can do it in a variety of ways that only take up a small amount of space: you can insulate the wall, put in a layer of sound insulating board (homosote for example), by hanging sheetrock on resilient channels, or with a sheetrock soundboard, which uses a special elastomeric material between two thin sheets.3
Notes
1: one idea is to allow up to a 4" incursion into the setback provided that the incursion consists of insulation beyond code.
There is still some controversy about how much insulation is necessary for moderate climates, and to a large degree its a tradeoff between up-front construction costs and long-term energy costs.
2: admittedly there is no standard thick wall construction technique, the most common techniques are double walls with one load bearing wall and single 2x6 or 2x8 walls with a foam skin.
3: alas, the material is quite pricey. You can also buy the material itself, which comes in tubes and is applied with a caulking gun. The only product I know of is "green glue".