I can't speak to cancer specifically, but in other disease areas there are mice bred specifically in to be susceptible to a disease. There are also ways to induce a disease, or a disease like state.

Knockout mice are given something that stops expression of a specific gene which can induce the disease like state.

I've also seen mice getting a kidney removed, which simulates failing kidneys in chronic kidney disease.

To simulate type 1 diabetes, you can treat a mouse with streptozotocin, which kills the islet cells that make insulin.

It can also be as simple as changing diet, to induce obesity and the associated comorbid diseases you can give mice what's know as a fast food diet, which is just their standard chow supplemented with a massive amount of fat.