The post-1962 cars running around with old plates on them have probably been registered continuously in California since they were new. As I understand the law, you get plates when you register a car and can keep them as long as it remains registered, so if you registered your Healey in 1965 and kept the registration up, you can still drive around with the plates issued in 1965. If, however, you restored an unregistered car, or moved from out-of-state with a restored car, you can only get YOM (Year of Manufacture Plates) if it is 1962 or older.
I make no apologies for California's Department of Morons and Vandals. Most workers there straddle the fine line separating animal an vegetable. My first choice for a plate (before I found YOM's) for my TF was "FWK-TF" it got rejected as being obscene (I already have FWK-S4 and FWK-Z3). My 90 year-old mother's ID kept getting turned dwon because they had her birthdate and social security number wrong. I finally got her an ID by calling Sacramento and telling them if they didn't want to recognize her as a resident, it was OK, because as a non-resident, she wouldn't be paying taxes here. The ID came the next day, but was marked "M" next to "sex". Not only does the DMV have trouble telling a "3" form a "5" but they apparently can't tell a woman form a man (and no, this wasn't the San Francisco office).