Reminder: If you switch cars or buy a new one – Call. Your. Insurance. Company. Don’t rely on the dealer

"We'll handle it with the insurance"

Often they won't. They just copy the information from your current insurance card and file the paperwork. Some salesmen are more concerned about the sale than the vehicle being protected.

But you have to have insurance to get the vehicle registered, right?

See above. They can just request or transfer the registration with a policy number, company, and effective date. Not necessarily an insurance card WITH the requested information.

But I need full coverage to get financing, right?

Yes, but again some lazy finance reps will just enter in your policy information, and probably have you sign a form saying you "have X deductible for collision/comprehensive" which has you again accepting the responsibility. Usually the prior vehicle's coverage declarations will be enough to proceed. (And stop calling it full coverage, or at least expecting the term actually means anything.)

But my insurance coverage will cover it anyway, right?

Not if it's not the company's error. Some companies will give you 30 days to notify them – e.g. if you have a wreck with the new vehicle on day 20, the most broad coverage on the prior car would extend to this accident. At day 31, that new car is as good as uninsured, and whether you have a wreck, hit-and-run, or even a nature incident you'll be stuck with the bill.

So is it ok as long as I call close to the 30 day mark?

Not necessarily. If you have a liability only car now, and are buying a car with collision/comprehensive or other coverage, the "extension" will only treat the new car as a liability only car until you notify the company of the swap and of the need for the additional coverage. If you're at fault your company would pay for the damage caused to the other party, but you'd be liable to the finance company. Get ready to pay through the nose for awhile (while you're trying to pay for a replacement too!).

Why should I have to pay? It's the dealer's fault! (Or the finance company, or the state, or the stars, or Trump, or Harris, or whoever…)

Nope, it's on you as the policyholder. You are required to notify your insurance company of any vehicle changes. If you handle it yourself, you'll be ok. If the agent doesn't process it correctly (and you have documentation that you requested it), then it'll be on his errors and omissions insurance.