"Sharks don't get cancer!" is a factoid that has burnt its way into the public consciousness to such an extent that nothing short of a cultural lobotomy can erase it. The perceived immunity of this predator to cancer has led to its slaughter in order to harvest its allegedly curative cartilage; not only is this no good for sharks, it's no good for humans either.
Sharks do get cancer – indeed, pretty much all complex multicellular organisms do, from dogs to elephants.I've never heard this one and wish I never had. Sure, it's possible to pass off the phrase as a hypothesis, but a simple search of PubMed unearths examples of sharks getting melanoma, liver and testicular cancer, as well as reports of many more in the Registry of Tumors in Lower Animals.
Being naive to this rumor, I originally thought it was related to the current media frenzy around "Sharknado", but no, the shark-cartilage/liver-oil-panacea meme has been floating around for over twenty years.
Here's a more comprehensive article at the Journal of the National Cancer Institute about the issues surrounding shark cartilage, its use as an anti-cancer agent in clinical trials, and the companies that sell dietary supplements fabricated from shark tissues. Even then (in 2005), it didn't seem like shark cartilage had a fighting chance against data it wasn't really useful in a clinical setting.