Confirmed with CSR. CLI throught the Aemx website will NEVER result in hard pull on your credit report. They are all soft pulls, no matter you get approval or not. {Edit} Some reported a hard pull from AMEX, especially if instant response was not given, the 7-10 business day response provided instead
At least one user reported AMEX FR as a result of CLI on crack play, with about 6 weeks since the original CLI and 2 weeks since the subsequent CLI
Credit Line Increase through AMEX website:
People are reporting success getting large increases instantly approved (like from 500 to 24,900), getting large CL instantly approved (34-35k). The only information asked is the CL you'd like to have and household income. Applies both to personal and business AMEX Cards (business cards also require business revenue).
The increases also available to new accounts (some report 3 days since receiving the card). Side benefit to getting high LOC on new cards - any promo offer running would be much more lucrative. This includes 0% for purchases only, as the current AMEX BT offers have no fee and offer the same rate as for purchases (but one would need to clarify if the standard or promotional purchase rate would apply for BTs).
Reasons to increase LOC include increasing your total available credit and therefore decreasing utilization, improving your credit score; also taking advantage of any BT offers; plus having large credit lines showing on your credit report may induce other issuers into giving you higher lines as well, which may be useful to futher lower your utilization and for consolidation during future app-o-ramas.
Problems people have been having. Decision is not instant (7-10 days) and yet to be made known, decision to decline the request. In the case of non-instant decision some report the CLI going through as confirmed by the phone CLI line call, as well as seeing the requested increase online at a later date.
Why this is not for the faint-hearted: DREADED AMEX FINANCIAL REVIEW (FR). Definitely do not do CLI if you can't afford to lose ALL your AMEX accounts (members have reported such negative actions in other threads). Most definitely do not supply income figures you can not substantiate - not only this would certainly result in losing all your AMEX accounts if you get FR, but also supplying false information on application for credit (and that is what it is) is a FELONY in some states.
Why do so many people ask for 24,900 credit lines? Well, some who requested more have been reporting getting approvals to $25k, but further approval was contingent on supplying tax return and other asset information (and perhaps resulting in FR and closure of accounts if requested documents are not provided).
If you do decide to take a plunge, do review this thread as well as AMEX FR thread (linked in this thread), and please play responsibly 
Q: How would one know online the account(s) was under Financial Review?
A: According to fotomaniac, "When you login it says 'Charging privileges suspended.' "
Another rule seems to be - if you carry a balance, you can only increase your CL up to $6,000. You can do this every month.