So, I managed to find a temporary solution:
It seems that when I play over wifi instead of wired it happens more frequently. My hypothesis is that it has something to do with the stability of your internet connection. My PS4 seems to try to connect to PlayStation Network and when the connection is not stable enough the connection is refused and it says unsuccessful connecting to PSN (You can check your connection to PSN by going to Settings -> Network -> Test Internet Connection). So although connection to internet is successful, connection to PlayStation Network is not.
What helps for me is:
- (Using wired connection to internet is preferable, but the fix works for wifi as well)
- Open Apex Legends -> start queuing for a match (says part not ready)
- Disconnect my router from electricity so it turns off
- Leave it turned off for a few seconds (to make sure it disconnects from the internet)
- Reconnect the router
- Apex Legends now says it lost connection, click Retry and queue again (party not ready should not appear now)
- You can check in the settings and it should say now PlayStation Network sign-in successful
It usually takes only one try but I have had to do it two times in a row a few times.
And I should also say the party not ready glitch sometimes returns the next day (and then it's okay two weeks in a row, so quite random).
Also I do not know why but it works better if you start queuing for a match before you disconnect the router (or it does not and I just got luck/unlucky).