Two issues to take into accoun to though.

1: Big banks contract out their services to smaller banks. Since smaller banks/credit unions don't have the resources to tooer the large varity of products that big banks do the smaller ones can just sell the big bank service with small bank's name on it.

2: Big banks have to adhere to more regulations when it comes to disaster recovery and such. The best example is NOLA, there were people with small banks there who lost everything and had no access to funds because the small bank was under water too. I don't know how it was all sorted out in the end, the news never mentioned since it probably happened a lot later.

Now the disclaimer, I work for a big bank, but the only services I use are the basic accounts that are free to anyone using direct deposit. I also have accounts at a smaller bank as well, gives me some redundancy.