So while the number of failed companies is not a precise predictor of the depth of a depression, it should give the right ballpark number. Since we can deduce company failure rates by their bond spreads, one can therefore estimate market-implied severity of a recession.
From Financial Times:
US investment-grade corporate bond prices, for example, imply a cumulative default rate of 36 per cent over five years, assuming a typical recovery of 40 cents in the dollar, according to analysts at Morgan Stanley. This is more than 7.5 times higher than the worst default rate in any previous five-year period.5 year default rate is not horribly useful, since a lot can change in 5 years. So let's convert it to a more relevant metric of roughly 8-9% of defaults per year. I think we can safely assume that the rate at which new mid-size companies are created goes down to roughly 1% from the typical 2-4% in 'normal' years. This means that within the next 2 years (typical time-frame for the worst part of recession) we are looking at the number of investment grade companies in US to shrink by roughly 15%.
So the market is pretty pessimistic and investors seem to project a recession with GDP decline in the ballpark of 10-20%, which puts it squarely in the depression camp.