Well, I found this
Overall roster limits have been expanded from 80 to 90 players per club, the NFL announced yesterday. This will provide teams with greater roster flexibility and allow more players to be developed and evaluated.
The 90-player roster limit, which goes into effect today at 4:00 PM ET, includes players on the active, inactive, practice squad, and exempt lists, and the following reserve lists: injured, physically unable to perform, non-football illness, non-football injury, suspended (for less than one year), future, drafted-unsigned, exclusive rights, first refusal rights, unrestricted free agents with an individually negotiated right of first refusal, franchise, and transition.
The first roster reduction is to 75 players on the active list and will occur at 4:00 PM ET on the Tuesday, August 28 following preseason week 3. The final roster reduction is to 53 players on the active list and will occur at 9:00 PM ET on Friday, August 31 of preseason week 4.
So the initial roster is active + all those = 90 players. Active has to be cut to 75 and then 53, but not sure how many spots that leaves for suspended players and the like. Either way, I can't find anything that says unsigned, suspended players serve out their suspensions while unemployed. It would make more sense if they could, but I just can't find anything to confirm that. I had heard that this was not the case, but also cannot confirm that. Seems like they categorize players suspended for less than a year differently, maybe?
EDIT - OK, also found this piece
from Mike Florio, which sheds a little more light on it:
Now that free-agent receiver Chad Johnson has pleaded no contest to charges of domestic violence, Johnson is subject to discipline under the personal-conduct policy.
But a source with knowledge of the procedure explains to PFT that Johnson likely won’t be disciplined until he returns to the NFL.
That said, if a team is interested in signing him and wants to know if he is facing a suspension, a decision could be made prior to Johnson signing a contract.
If Johnson had been issued a suspension before he was cut by the Dolphins, he could have served the suspension while unemployed. Not that he’s unemployed but has yet to be officially suspended, the league apparently isn’t inclined to suspend him until he has a team.
So do we know if Hartmann's eight game suspension came down after he was cut by the Texans, or before? I think the three game suspension came while he was still with the team, but wasn't sure about the latter. Also, it looks like the league uses the suspension mechanism to blackball players at times.