Is there actually a cave entrance on every screen?

Page 31/32 of the manual for The Legend of Zelda for NES says

Legend has it that there are caves in every possible place above ground. Link is bound to find them as long as he has enough power. Get the idea?

I knew there were a lot of secrets hidden within the game, but I never realized there is supposedly a cave or secret on every screen? Is this true?

And as a bonus, if it is true, could someone link to a map that shows them all? :) Thanks.

No, there isn't a secret entrance on every screen. Additionally, not all of them are accessible right from the get-go. Some require items that you collect through game play, like bombs.

An overworld map, as well as all hidden locations, and where to find what (like free rupees, potion shops, gambling spots etc) can be found here.

There isn't a cave on every screen, but there is adventure to be found on every screen! The number of secret items/areas/caves from BOTH quests combined IS close enough to fill 1 entire overworld in LOZ, but that's it. 16x8=128 screens.

Combining both quests' all found overworld items, cave entrances, or warp zones, you get 8 total warps, 10 total heart containers, 2 letters, 4 potion locations, 16 dungeons(level 1 in same spot), 5 sword locations(starting sword in the same spot), 8 gambling locations, 20 shops(some contain free items like the braclet), 20 pay-for-repairs, 28 free rupie locations, 2 blue ring locations, 4 fairy locations. If i missed anything then add it to my total of 119 screens that have items. I might have missed some stuff, such as counting the lost woods(you have to take a specific path, so kind of like a secret), or the waterfall climb(up 5 screens). But as you can see, it's pretty close!