I've had two successful AmEx CLIs, both done when my balance was at $0.
TU FICO: 800 (2/1/14) | CK Score: 802 (2/1/14) | CS Score: 805 (2/1/14)
J.P. Morgan Palladium ($250k) | AmEx Platinum (NPSL) | AmEx SPG Personal/Business ($50k/$50k) | Citi Executive AAdvantage WEMC ($50k) | Citi Dividend WEMC ($50k) | Chase Sapphire Preferred VS ($50k) | Chase Ink Bold WEMC ($50k Flex) | Chase Ink Plus WEMC ($25k) | Chase Freedom VS ($25k) | Chase Freedom WMC ($25k) | Chase MileagePlus Explorer ($25k) | Chase Southwest RR Plus Business/Personal ($15k/$15k) | Barclays US Airways ($25k) | Barclays Hawaiian Airlines ($25k) | BofA Alaska Airlines ($10k) | Lexus Financial Services ($30k) | Mercedes-Benz Financial Services ($50k)