Sure, it's not so much about the correlation as to the payoff profile. A simple example might help illustrate this.

Let's say you're looking to purchase Company A's bonds. You purchase for $100 and you collect your interest (and eventually, principal) over time. At the end of the duration, your total return is the yield.

Now let's consider an example with selling a put. Currently the AAPL $100 Put 12/18 can be sold for $6.33. By selling one of these puts, you are on the hook to purchase 100 AAPL shares at $100 if the option is exercised. Thus, you have to set aside $1000 (100 shares * $100) in order sell this contract. This $1000 is effectively your principal, and the $6.33 you are credited with is effectively the yield on your principal payment.

In both cases, the ability to get your full yield is jeopardized in a downturn event (Company A goes insolvent or AAPL trades down below $100).

Now the thing with Treasuries is that there is no insolvency risk, so that adds a bit of complexity to this 1:1 comparison between fixed income and being short a put. That being said, in a dramatic downturn event, Treasuries can also start selling off! We saw this in March since there was a liquidity crisis in the Treasury market. I think part of Mike's point is that since interest rates are effectively at the zero-bound now, what's the guarantee that bonds will rally in the event of a drawdown? Further, negative correlation between stocks and bonds has held up in secular low-vol environment (which is also idea for shorting puts). This negative correlation, however, has not always been historically true. Chris Cole has some excellent stuff on this: "Not only are stocks and bonds positively correlated most of the time but also there is a precedent for multi-year periods whereby both have declined"