Thanks Nidhin,
We did check the save username and password the first day when the connection was allowed.
In the C:\Users\[username]\AppData\Local\Temp\sdedc_PostgreSQL.log file, the following error is reported:
CAN'T OPEN INSTANCE: sde:postgresql.
Spatial Engine Connection Failed (-9).
Cannot Get Access to Instance sde:postgresql
The most "odd" part of this to me is that the connection worked one day for many users, but then failed to connect the next day. I have been able to maintain access to the postgresql server throughout, but other users have lost it.