There's a whole thread on this you might want to check out titled "The Definitive 3X Amex CLI Guide" or something to that effect but it's a long read, so I'll sum it up for you.
You're eligible at 60 days and then upon approval at 180 days. If you're declined on a request and you're in the eligible period you have to wait 90 days. You'll need to wait until the day after, so you'll see this referenced as 61, 181, and 91 days on MyFico.
If you're declined and you aren't eligible, your eligible date doesn't change.
You can request anytime. First eligibility is 61 days after opening the card. If you are denied a CLI at any time due to credit-related reasons, there is a 91 day waiting period before you're eligible. If you are given an increase, you must wait 181 days before you can be increased again. If you're denied simply because you pulled the trigger too early, it does not reset the clock - you just wait however many additional days it takes to reach your next eligible date.
ETA - beat me to it 
@Anonymous wrote:You can request anytime. First eligibility is 61 days after opening the card. If you are denied a CLI at any time due to credit-related reasons, there is a 91 day waiting period before you're eligible. If you are given an increase, you must wait 181 days before you can be increased again. If you're denied simply because you pulled the trigger too early, it does not reset the clock - you just wait however many additional days it takes to reach your next eligible date.
ETA - beat me to it
Haha but you explained it better!