I recently moved into a condo that had been empty for several years. Maybe about 6-8 weeks in the toilet clogged and overflowed. The water went downstairs and I had someone come in to rod it to make sure it was clear. Everything seemed find other the slow drain I'd been dealing with in the bathroom sink before the toilet flooded. Another 6-8 weeks go by and my downstairs neighbors now say that whenever we flush their bathroom smells like waste. Of course, the HOA is asking if it's possible the pipes were damaged when we were making sure the toilet drain was clear. Again the smell has gotten worse but comes with flushing upstairs ( they can hear the water coming down when we use the bathroom, I presume, because I can hear with the bathroom is in use above me). The only other thing of note is before the sink started draining slowly I had snaked the tub that was NOT flowing and it cleared the tub but then the sink started slowing. I tried to unclog my sink two days ago and now it seems worse then ever instead. Don't know if there is any relationship. What is the most likely source of the smell downstairs?

When snaking a drain makes other fixtures worse, then you have not adequately addressed the blockage, just moved it. The smell is likely from other dried up fixture traps or even unseen leaks that allow the sewer smell to escape. Maybe there are unoccupied apartments within your proximity? I would have to do an assessment of your entire building plumbing issues to be specific or certain.

You really don't know if other tenants are having issues like yours do you?