The whole intent of that is to prevent friends and same accounts from facing each other. Sometimes same networks are on at the same time. There should usually still be queues available for teams.
What does this mean?

To the developers of 4 Player Chess.
I feel like my privacy is being invaded. I do not know who that other player is or if it's a false positive. I am afraid either my PC is hacked and this person is stalking me or someone in my family plays 4PC which is very unlikely. I do not know their business on the internet but if this is the case I'd be very surprised. For a long time, I've had an uneasy feeling my internet was being watched by a 3rd party. I was hoping you could look up and tell me the username of that other account, or at least confirm for me that it's a false positive and that there is no other player on my network. Thanks. Best.

I am not a dev, just a 4PC sheriff but:
1. We cannot disclose the username of a player “in your network” UNLESS we think you are a secondary account of that user, and we cannot do so publicly.
2. We cannot even know who that user was that was in the queue you tried to enter. If Thing1, Thing2, and Thing3 share the same network and you are Thing1, we could not even guess who was in the queue even if Thing2 and Thing3 are both online at the same time.
3. Same network doesn’t necessarily mean hacker. It could be someone at your school, cafe, or even across the world using the same provider.
4. Same network doesn’t apply to the one you are using right at the exact moment and is not limited to just one location. If you were on the site from a certain public location on December 21, 2019, and a random user was also on the site from that same public location but on February 21, 2020, we consider that a network match and prevent you and that player from playing rated games together just in case you and that person are prearranged teamers, etc.
IN SHORT, the whole purpose of this setting is to prevent cheating.

I am not a dev, just a 4PC sheriff but:
1. We cannot disclose the username of a player “in your network” UNLESS we think you are a secondary account of that user, and we cannot do so publicly.
2. We cannot even know who that user was that was in the queue you tried to enter. If Thing1, Thing2, and Thing3 share the same network and you are Thing1, we could not even guess who was in the queue even if Thing2 and Thing3 are both online at the same time.
3. Same network doesn’t necessarily mean hacker. It could be someone at your school, cafe, or even across the world using the same provider.
4. Same network doesn’t apply to the one you are using right at the exact moment and is not limited to just one location. If you were on the site from a certain public location on December 21, 2019, and a random user was also on the site from that same public location but on February 21, 2020, we consider that a network match and prevent you and that player from playing rated games together just in case you and that person are prearranged teamers, etc.
IN SHORT, the whole purpose of this setting is to prevent cheating.
So if another random player and I used the same network (for example - VPN) at different times in the past, and we both tried to join the same game queue, the error message in the picture will be shown?
No one in my network (home) plays Chess let alone 4 Player Chess. What is this? Could it be a false positive?