Well, that's how some people do it. They shop for a car with 11 months left on the license plate stickers, and then just drive it until the sticker expires. If they behave themselves in traffic, and nobody else runs into them, and all the lights on the car work, they can often get away with it.
These are -- generally speaking -- the folks who drive without insurance, and that's a crime.
And the law in California allows you only ten days to start the registration process when you buy a car.
Also, the seller is supposed to report the sale to the DMV within five days.
The authorities have all this information on computers, so any law enforcement officer can simply run a check on the license number, and know all about the car in a minute or less -- including whether the car has liability insurance. Without insurance, the car's registration is automatically suspended, and you will be seeing a bright red light behind you.
Now.... I've been talking about private transactions. If you buy the car from a dealer, then the car will have been smogged, and the dealer will give you the Temporary Permit. I do not know how it works with the insurance. Generally, insurance companies give you maybe as much as 30 days to add the new car to your policy -- IF you already have a car insured with them. But you do not have an existing policy, and I do not know how the dealer would... eh... deal with that. I suppose I could telephone a couple of dealers and ask.
But you will pay more for a car at a dealer than from a private individual. And few dealers have cars as low as $1.000,-.
Hope this helps.