Can you block noise by adding mass?That depends on the situation. What kind of noise? How does the noise travel? What materials will you use, and how will you apply them?
This 2008 article from "Sound on Sound," explains that sound "cannot be destroyed: it can only be converted into another form of energy." While some sounds are airborne and can easily be blocked, you might still hear lower-frequency sounds passing through on the other side. This is because the energy from sound waves makes the walls vibrate, and this creates more energy.
Does adding mass help? Yes. Will mass solve all your noise issues? Possibly not. You might also need to address other characteristics of the space by adding sound absorption inside the rooms, or by decoupling the floors, ceilings, or walls from the structure to keep the sound (energy) from passing into it.
Let's look at a couple of examples.
Example 1: Interior doorsMany American homes feature hollow-core interior doors. These doors have a cardboard "honeycomb" grid sandwiched inside two thin slabs of particle board, and they're not effective at all in blocking sound. You can replace these hollow doors with solid wood -- adding more mass. Or you can replace them with a solid-core door made from varied materials, where the core might be filled with materials like mineral core or wheat straw.
Although the mass in a solid wood door will block noise better than a hollow-core door, the mass plus absorption properties in the third option -- a solid-core door -- might be even more effective. Solid-core doors are typically fire-rated, as well. This article from The Spruce gives a brief overview of the three categories of door options.
Example 2: ConcreteIf you've ever spent time in a warehouse or a converted loft space, you know that the density of the concrete walls and floors can block some airborne noises from passing through them. However, concrete can still conduct impact noise, and -- as this article from Acoustical Solutions states -- some concrete has air cavities in it that might allow sound to pass through it. Additionally, the walls themselves might reflect noise around the room.
Conclusion:In general, when you address noise issues in a space, you have consider more than just adding mass to it. You also need to consider sound absorption, and you might need to look at decoupling some surfaces from the structure, or to create flexibility. For example, a viscous product like Green Glue remains flexible after you apply it between two layers of drywall, and it converts noise energy into a small amount of heat.
Of course, the Berlin Wall isn't related to the topic of noise control. But it's a great example of how creating a lot of mass didn't stop energy from passing through it.
[Updated in 2018]