I had this issue on my PC (running 4 accounts), with one exception. On one account, if it didn't get stuck on "Playing", it would log me into another of my accounts instead. I deleted all the Saved Credentials from the Launchpad screen and then manually logged in the 4 accounts, saving the info as I went. Everything has been fine since then.
I have had this issue a few times. From what I can tell, sometimes the launchpad seems to lag when switching accounts, and if you click play on account2 immediately after account2 gets shown on the launchpad, account1 that was previously selected gets recorded in memory as the account you just opened eq with and locks you out from logging accout1 in. Since I noticed the bug was corresponding to the lag, I have been watching for the account I want to log in to have the name pop, then count to 5 before hitting play. While doing this I haven't encountered the isssue, but when I am rushing to get several accounts on, I have. I would also say that this might not be something a lot of people encounter as if you log accounts on in a certain order, it would only really be seen logging the final account on when you rushed in getting the first account on as all the middle accounts would just be tied to an account that is already on.
I have noticed the "lag" part of it on ocassion too - but now, if I just leave the patcher sitting there waiting for it to "catch up" and find the information for the account I'm trying to log in, the patcher just times out and closes.