If I were put on a PIP, I'd start looking for another job right away, it is usually the first step in firing someone and the PIP is their way of documenting why you were terminated. I think cases of getting yourself back in good standing after a PIP are rare. They are NOT obligated to give you severance if you are let go due to a PIP. I am not sure about COBRA.
In fact, US companies are not obligated to give severance for restructuring layoffs either, but most do in order to maintain reputation. But being let got due to a restructuring layoff is not the same thing as being let go due to a PIP.
I do know one person who was on a PIP and was part of a big layoff, she lucked out frankly, because she was able to say she was let go due to restructuring and not due to her poor performance.