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Hi, cofounder of Simple here. We work with partner banks who hold on to deposits. For all intents and purposes, you never have to think about who that partner bank is, but if you ever need to contact them directly, you'll know where your funds are. You might find our FAQ informative: https://www.simple.com/faq/ .
Answering etherealG's question here because I can't reply to him below: Yes, you'll know what institution your funds are at. When you get your Simple card, it'll say "issued by {Bank Name}" on the back.
(Also, what bank do you currently work with? What will the first round of cards say?)