Yes, this is true. I have never beaten the oringinal Super Mario Bros. on the NES/Famicom. To be fair, I haven't really tried to beat the game yet. I usually just play until the first warp pipe then when the nostalgia wears off, I shut off the game.
When I was growing up, the original Famicom that we had broke before I really got a chance to play on it. So my folks got me one of the NES clones called "Super Family Computer" which is just a NES in a SNES body. The controllers had four face buttons on it which X and Y were turbo buttons of A and B. When they got me that, the old games we used to have were either sold or already given away so my mom bought me one game and it was Mega Man 6. Rockman 6 to be exact. One of my neighbors (he was around 5 years older than me) found out that I got a new NES and gave me a sack, a literal sack, filled with his old NES cartridges. Among them were Rockman's 1-4, I got 5 from another neighbor. A ton of other games were also in there such as Super Mario Bros. 1-3, Mike Tyson's Punch-Out!, Spy vs Spy, Battle City, so on and so forth.
I was in the situation where I had quite the number of games to choose from and wasn't forced to just grind away at this one game that I owned. If I lost interest in a game, I would just simply pop in another one. I really enjoyed Super Mario Bros. back in the day but I had been exposed to a lot of more refined games before I really got to play it. Aside from the shortcut pipe and hidden 1-Up in world 1-1 and the warp pipes in world 1-2, I don't really know anymore secrets in the original Super Mario Bros. One day I'll sit down and try to beat the game legit, no warp pipes. But for now, here's a vid of me playing SMB while I messed around with a NES emulator.
In my defense I did beat Super Mario Land on the original GameBoy.