The answer is more like "It depends on the protein, and the renaturation (or refolding) process." There are a lot of factors that contribute to an individual protein's ability to refold, including size, sequence, secondary structure, amount and type of inter-amino acid links like disulfide bonds, number of subunits, the presence of chaperones/heat shock proteins, and, yes, how it was denatured in the first place (sorry, I couldn't resist). Smaller proteins will refold more easily than larger ones. Hydrophilic proteins tend to refold better than more hydrophobic ones, especially membrane-bound proteins. Multi-subunit proteins/complexes tend to need some help to properly reassemble. Adding in chaperones/heat shock proteins will almost certainly help the process along for all but the smallest and hardiest samples, and will give you better results than just dialyzing all the salts/detergents/chaotropic agents away into PBS. Finally, if you denature by boiling in Laemmli buffer, you're going to have a very difficult time refolding most things, while going from Guanidine HCl to PBS isn't always that bad, depending on what you're looking at.
So, unfortunately the answer is "it depends." You have to remember that the intracellular environment is very complex, and is designed to correctly fold proteins after they're made, and break down misfolded proteins before they can aggregate or cause other damage. There are hundreds of thousands of proteins just in humans, so it is rather difficult to make blanket statements, but for the sake of resistance to damage, many smaller, single-subunit proteins can likely be at least partly to mostly refolded and regain some or all of their original activity. As complexity grows, however, the likelihood of successfully regaining function decreases.