Players are able to run two or more instances of Minecraft on a single computer, and were able to login successfully in the instances of Minecraft running on that computer. (Hence, the environment is generic.) I'll assume this behavior is correct.

Since they are able to run two or more instances of Minecraft, they have the ability to join a server with matching account names. The first instance of Minecraft is named as A, the second instance is named as B, and the server that both instances, A and B, wants to log into is named as S.

Currently, it goes like this:

1. A logs in to S.
2. B logs in to S.
3. Since B's name is the same as A's, S kicks A, and lets B in.

What it should really be is this:

1. A logs in to S.
2. B logs in to S.
3. Since B's name is the same as A's, S checks the IP addresses of both A and B.
4. S kicks B, and keep A in, because their IP addresses are a match.

OR in the case of different users, it should use its old method:

1. S kicks A, and let B in, because their IP addresses are not a match.

This affects only a minority of the Minecraft players out there, but this is still an issue that exists since Alpha 1.1_02, meaning that it affects all versions of Minecraft. This issue was not visible to Mojang in GetSatisfaction due to low vote counts and low comments.