For what it's worth, I often find the studio is slow and spottily-responsive when working on my projects late at night - around mindight/1am-ish, UK time. It's generally OK before then (between say 8 and 12 - I don't often do much in the studio at other times of day or night so can't comment on that).
But by this I just mean that when making a move it takes forever to respond, and sometimes gives be the "are you connected to the server" red banner - ironically quite often the move registers and the interface updates just after that (but not always, sometimes I have to resend the move, not infrequently several times in a row). That's different from seeing a 504 error, which I've never noticed myself. So probably not the same thing at all.
Having said that, I'm concerned about @Een's responses, that a dev could basically bring down the whole Studio with malicious code (whether deliberately or accidentally malicious). Granted hackers probably won't bother with BGA studio, but then it seems that no "hacking" is actually involved - you would just have to register an account, start a project and stick in a "while (true) { }" in setUpNewGame, start a game and walk away. Granted it's probably easy for admins to spot and shut down obvious stuff like that but it's a bit worrying that it's so easy to kill the server.