I think there are too many variables involved to be able to come up with a black-and-white answer.
Just thinking in terms of my current cards, some of them *never* get CLIs while others get auto-CLIs and CLIs I request, even though their utilization is very low. And historically very low. With others, I get auto- and/or requested CLIs with a lot of usage. In other words, just with my handful of cards, it varies! I'm thinking there's a lot of individual factors involved in their decision-making process, so what works well for me may not work well for you.
Something I read quite often here is that you have to really *use* your card before its bank will even think about increasing your credit limit. But I'm living proof that that's not necessarily true. For my cards. With my factors.
FWIW, none of my three Cap1 cards has had a CLI in years; I don't use any of them heavily. Most of my other cards, including Amex, BofA, Citi, and Discover do give me CLIs, and my usage with each of them varies from nonexistent to very heavy. Usually, more on the nonexistent end of the scale. I've tried, and failed, to find any rhyme or reason to it! For example, I've tried asking for a CLI on a specific card when its utilization was much higher than usual versus asking when it had a zero balance--the results didn't vary much!
I hope someone else can chime in with something that's actually useful. 
Which bank are you referring to? There are known banks that are more generous than others. CU's lead the pack it seems when it comes to CLI's.
As others have said, all over the map. Speaking generically, steady use and a good payment history probably helps. But one of my cards (Coastal) has grown nicely with almost no use. Other cards (Apple and Discover) did not grow much despite moderate to heavy use.
@CreditCamelMT wrote:What utilization do you need for banks to keep giving CLIs? Assuming good scores and PIF, how much usage do cards need to continue to grow?
Any DPs on what metrics they look at and optimal values? I'm thinking maybe some combo of Average util, Peak util, and Percentage of income.
Any thoughts and guidance appreciated!
Every issuer has a different policy, and I would say most have no consistent policy at all. The closest things I can think of as sources for some kind of guidance are:
(a) experience with Chase seems to suggest that heavy use coupled with quick repayment sometimes catches their eye, especially at the 6-month point, for an auto CLI
(b) Barclays once posted an article saying that the people who had the best chance for a CLI were those who spent a good percentage, but not too much; the chart they published with it omitted the actual percentages, but it looked to me like they were saying between 10 and 30%. However, in practice, they did not seem to follow anything like that, and were pretty stingy on CLI's
To be honest, I think it's an exercise in futility to try to figure these institutions out.
@CreditCamelMT wrote:What utilization do you need for banks to keep giving CLIs?
Assuming good scores and PIF ... Average util, Peak util, Percentage of income.
Welcome to My Fico Forums, @CreditCamelMT.
As you may see already from some great replies, it's a complex topic. Your first question can't really be answered at all since (1) it's an oversimplification that assumes utilization is a always a significant controlling variable of credit limit (which it may or may not be); and (2) when it does apply, the utilization needed depends on both the lender, relationship, spending, and credit profile. So your question, when applicable, is only focused on (ONE) of those four factors.
I have similar interests to yours in wanting to understand what drives credit limits, both SLs as well as CLIs and high CLs as a whole. Besides learning about the topic as well as try to grow my own limits, I wanted to share what I had learned with others. I've posted several longer threads about this before. If you've like to read them, start at >>this link<< and follow the other links to another few prior postings.
You've mentioned utilization and you've mentioned FICO score. I see FICO as more of a "gatekeeper" role for limits. A low one will often keep limits low (regardless of utilization) but a high one won't necessarily get high limits without other variables being in-line. The main issues with growing limits are (1) time (2) money ... or both. The money may take the form of income; debt that is in line with or low relative to that income; spend-and-pay (utilization) which may be related to personal income or reimbursed business spend; and assets invested with a financial institution. Someone that is a high net worth/ultra high net worth client (HNW/UHNW) may attain high limits or high CLIs without spending based on the assets invested. Absent assets, most lenders will assign SLs and limit CLs based on ability to repay, i.e. income. So utilization on a card may only drive a limit to a certain point before you've passed their comfort level, as your reference to percentage of income acknowledges. Each lender will set their own percentage in those cases and that percentage may vary based on overall profile.
Don't discount the impact of debt-to-income either. Someone can be making a high income but if they are carrying large student loan debt or a mortgage that is a heavy burden, opportunities for higher CLs may be constrained.
Your follow-on post above mentioned file age, lender underwriting variations, and credit-seeking behavior patterns which further muddy the waters. Those are all also important variables. It's impossible to boil it all down to even vague general utilization ranges since it just depends on profile and circumstances.
For example, Navy Federal CU is an outlier in that they are more generous with builder/rebuilder profiles and they often give out limits higher than a profile would merit with other mainstream lenders. They will also give regular CLIs without significant spending on their cards. Many other lenders want to see some level of spending, but how much depends on who we're talking about. Some lenders will grow your limits with only modest use to extremely high levels if you're a loyal and responsible customer for long enough. Some of my cards have grown that way over decades. There are threads with data points on various lenders if you want to research specific ones.
Your strategy ("I'm specifically interested in high limit cards with top tier issuers Citi, Chase, BOA, and/or Amex -- don't want a wallet full of 1-5k cards that I'm shuffling spend around to avoid util penalties.") is very similar to what I have been seeking. You're on the right track since lender size, while not always a corollary relationship, is often related to CL potential. If high limits are a goal, picking the right lender is important. Some lenders may limit cards to $10K or $15K or $25K to limit their risk exposure. Large lenders have deep enough pockets that they will often feel comfortable extending more credit. (This isn't to say that there aren't also some small lenders with generous limits.) But as I mentioned, if the other factors such as income, FICO, debt, file age, etc are not in-order, CLIs will still be constrained with the large lenders, regardless of your utilization of their cards.
And as @Kforce pointed out, will pushed credit limits stay that high with normal spend? With some lenders, if you don't use it you lose it. Or sometimes they will be quick to cut limits if there's an economic downturn (such as during the pandemic) or if there is a hint of financial distress in your profile. In general, I've found that with most lenders, they've allowed my limits to remain in-tact with light-to-moderate usage as long as my FICO and overall profile have remained in good standing. This is another reason why I'd rather have fewer, higher limit cards since it would focus enough natural spending on each one more easily without having to put a card on utilization life-support. When a profile's number of cards exceed the budget to comfortably support them, they may stop growing and they become more of a hassle to manage. On the other hand, limiting cards gives the ones you most care about the opportunity to blossom. IMO, the gardening phrase we use on My Fico isn't about just abstaining from new accounts/HPs but also "weeding out" our less useful cards which allows those most treasured to reach their potential.
