Inquiries show up instantly, so Bank #3 will know that you made two previous applications that day (assuming they pull the same bureau). The advantage is that new accounts will not show up on your report yet, so there is less risk that the later banks will be worried about you opening tons of new credit lines since they can't see it, yet.

I did get denied on the Club Carlson due to "too many new account" - note accounts, not inquiries. I think I had five new cards in the last six months. I called recon and they asked why I had so many new accounts. I told them that I was simply doing my annual credit card reward realignment and that since I had some trips to Europe planned, the CC card would work well for me compared to other banks. I also offered to close/convert another old/unused US Bank card, which he noted and sent it off to a credit analyst. I ended up approved without having to close the other line.