I have been having this same problem, just started today. "Server timed out. Please try again. Error code: 60002.503.999". I have only used this game or had it installed on a single device (my phone), and now I can't get logged in, i just get the error code. I've spent way too much time building this city to just start over. If it can't be fixed, without losing progress, I'm out.
For what it's worth, I've seen an interesting issue on both this and the Simpsons Tapped Out app. Both will periodically say it looks like I've been playing on another device... but I haven't. A glitch, perhaps? IDK, but it doesn't happen often.