To think one would ever have three months savings in cash sounds insane to me.

In the USA unexpected expenses (beyond monthly savings) can completely wipe a three month savings in a heartbeat.

Add to that getting laid off during a recession and you’re in for real pain.

I guess it’s just been 12 years since people ever had to deal with a bad economy.

What if your car suddenly breaks down and you have to replace it? Is the extra income of investing, let's say, 3 months of expenses really worth it?

Let's say you get 3% in an instant savings account, and 3.5% in a three months' notice savings account. If you never touch that money until your retirement in 30 years, that would give you 0.5 months worth of expenses extra interest.

Compare that to the hassle, extra work to keep track of several savings account, and the added risk in the unlikely event that you do need the money quicker (eg. A broken car, an unexpected medical emergency, whatever). Surely the trade-off must than tip in favor of simply having a bit bigger emergency fund?

You could use that logic to say keep all your savings in cash all the time. What if your roof gets blown off and you suddenly need to replace it? What if your house burns down and you need to buy a new one? The relevant question to ask is when having an extra couple of months' expenses in instant access is going to make the marginal difference. IMO the window between "one months' expenses" and "six months' expenses" is pretty narrow, so it makes more sense to keep one month or so in cash or current account and then invest the rest at the best rates you can.

(e.g. I would say for most people a car probably costs more than three months' expenses, so having that much extra in instant access probably doesn't shift you from "can't buy a new car in cash" to "can buy a new car in cash". Realistically you're (hopefully) insured and you're quite likely to finance your replacement car.)

I've had an "instant access" savings account take 2 months to pay out my money - some kind of AML issue, supposedly. Conversely, a lot of "three months' notice" accounts aren't "you physically can't take money out with less than three months' notice" but rather "if you withdraw with less than three months' notice you sacrifice one year's interest" or similar. I don't find the "hassle of an extra account" to be significant when it's two accounts at the same bank under the same login, but if you do then I would seriously consider having only a notice-requiring savings account.