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Possibly... computers and manual review may take existing limits into account.
But one thing i can say for sure, the things you do that get you high credit limits will help you get other high credit limits. So once you have one, if you keep doing the same things, you are bound to get other higher limits.
But once you get high limits cards, you have to be conservative, keep INQ's low with only 1 or 2 in a 6 month period at most, UTL low (less than 30% on any one card and total of less than 30% as well, no lates or other baddies, or you will stop getting high limits and will see limits reduced. YMMV and high income and years of history with a lender may allow you to break these rules, no promises.
Congrats on your new credit lines, and it looks like you're definitely reaching your few cards/high limits goal
It has worked that way for me, though as others have said, it may vary depending on credit profile. I had only one card at a low limit for a long time (800), and they kept denying me CLIs, until I got fed up and applied for a 2nd card that gave me a 3000 SL. Then after a few months I called the 1st, had a frustrating conversation (they insisted it had to be HP) with still a denial. Then suddenly, I got an email a couple days later, a CLI making 1st card limit > 2nd card! I do believe that someone reviewed that call and SP, and seeing the 2nd card's limit/usage made them want me to use 1st card more.
My goal is like yours - I'm determined to stick with the 4 cards I have now for a looong time and just keep getting CLIs.