I don't think USB has a limit on the number of cards you can hold at one time. I might be wrong but, I believe they do have a total amount of credit they'll extend.
I have 4. Two Cash+, AG and AR. I also have a cousin card through Elan with the MCP.
US Bank is always sending me links in the app, emails and snail mail to apply for more.
I personally have 4 and I think I have seen someone that had 5 or 6. US Bank will usually limit you based on how much credit they are comfortable to hand out to you and how greedy you are. They are not fans of credit seeking so will keep you out if you try that. Conservative math is what they like. I will say getting any card from them is respectable and having more enters unicorn status. I have had lots of cases where I got a new card from US Bank and terminals would stall because the links in the back would not be active yet. Basically I would be the first person to need a link to US Bank from the merchant processor. It would cache it afterward.
@tcbofade wrote:Currently, I have three. Cash+, Platinum, and State Farm Good Neighbor.
Total CL = $35k.
I'm debating the shopper rewards card this summer.
Can I open a fourth account?
Please and thank you.
I have 3 personal and 1 business card with limits of 25k, 9.6k, 9.6k and 3k (Total 47.2k).
I have read here that they don't liketo give over 25k on one card without a HP so I put a hold on asking for a CLI on the highest CL on my Cash+. I am trying to build the others before subjecting myself to a hard pull.
@Titans84 wrote:I have 4. Two Cash+, AG and AR. I also have a cousin card through Elan with the MCP.
US Bank is always sending me links in the app, emails and snail mail to apply for more.
Curious - do they send you offers to apply for a 3rd Cash+?
@Wavester64 wrote:
@Titans84 wrote:I have 4. Two Cash+, AG and AR. I also have a cousin card through Elan with the MCP.
US Bank is always sending me links in the app, emails and snail mail to apply for more.
Curious - do they send you offers to apply for a 3rd Cash+?
I haven't received any offers on cards I currently have. My last approval was the AR in September. The last time an offer was sent to me for that card was snail mail from July. Nothing since being approved. I'm currently being targeted to apply to the Altitude Connect, Shoppers Cash Rewards and the State Farm Premium Cash (I have insurance with them). I also get targeted to apply to almost all the business cards.
Since US Bank has only 6 core cards, it should be difficult to find a person with 4+ cards but they may exist 
I have 3 core cards and one co-branded and they are sufficient for me. I guess, there should be no limitation on co-branded. If there would be a limitation on branded ones, you may get the equivalent of Cash+ from Elan and get the rest of the core cards from US Bank. Based on DPs, Elan also gives very generous CL vs US Bank.
Fico8: EX~EQ~TU~840 (12 month goal~850).