Hello, I've been running servers from my home internet for quite a while (many years) including dozens of minecraft servers in the past.

My friends and I were trying to start up a new playthrough on 1.13, using the same port settings we had been successfully using just a day prior to run a modded 1.7.10 server, when we found that he could not join our server. The server showed up, as it was ported correctly, but when he tried to connect, loading terrain would show up for 5-10 seconds (much longer than usual), and he would get in game for about a second before being disconnected with the message "Internal Exception: java.io.IOException: An existing connection was forcibly closed by the remote host".

We tried a lot of troubleshooting, checked to see if idle timeout settings had gone wrong (set to 0 , the default).

Nothing worked, so I figured we ought to see if 1.7.10 still worked- and it did without issue. We then went through the updates to see which update it was where he could no longer connect. Everything through 1.12 worked just fine- I tested some of the snapshots- 1.13-pre5, and they failed too. So I don't know exactly where this issue began, but it was somewhere between 1.12 and 1.13-pre5

Today I got the chance to try and log in on my laptop from a different connection, and found that I could not connect either- which confirms that it's server side and not just a problem on his end.

I've yet to find anyone with the exact same issue as me, but I did find some similar issues:

http://www.minecraftforge.net/forum/topic/65198-kicked-from-multi-player-server/

https://www.minecraftforum.net/forums/support/server-support-and/2917715-only-2-people-able-to-be-on-server-at-once

https://www.reddit.com/r/Minecraft/comments/93qc7g/ioexception_an_existing_connection_was_forcibly/

TLDR;

It is server side- people joining from the same internet connection as the host can join (myself, and my girlfriend were able to from my house), while anybody off of my internet could not (my friend in Texas and myself on a laptop not home).

This issue began to occur on a version somewhere between 1.12 and snapshot 1.13-pre5, and occurs on all issues after pre5.

Update-

It seems that the disconnect message is inconsistent too. Now that I'm on a separate internet I can test this more thoroughly than my friend wanted to. 

Messages I've received:

Timed out

Internal Exception: java.io.IOException: An existing connection was forcibly closed by the remote host

 one about a connection refused with java.io in there (forgot to screenshot it)