Here's a thread dedicated strictly to your question.
http://ficoforums.myfico.com/t5/Credit-Card-Applications/The-Definitive-Amex-3X-CLI-Guide/td-p/18110...But here's the summary:
- new accounts eligible for CLIs 61 days after account opening; requesting too early does not reset the clock
- if denied for reasons other than too early, it's a 91 day wait til you can try again; it's a 181 day wait if you are approved for an increase
- can only get one CLI for the entirety of your AMEX account every six months following approval so keep this in mind when choosing which card to get an increase for if you have multiple revolvers
- auto CLIs from AMEX do not reset the clock on customer-initiated
- said auto CLIs from AMEX are rare but happen