Here is how it worked for me.

I have a Long Term Disability policy from my employer. When I went off work, I was first put on Short Term Disability and then Long Term Disability receiving 2/3 of my salary tax free, which is essentially full pay. I went on Cobra, which I had to pay. My LTD company had me file for Social Security Disibility, which was granted. The LTD payments were offset with the SSDI, so I still recieve 2/3 pay but some comes from LTD and the rest SSDI.

After two years on SSDI (which went retro to the time I went off work), I started to receive Medicare.

Fed Cobra lasted 18 months, but when I was granted SSDI, Fed Cobra was extended to 29 months. After that, California Cobra picked up so that Cobra lasted 3 years. If I was to do it again, I wouldn't use Cobra, I would get a plan through the Affordable Care Act. It is less expensive, with better coverage.