The "increase" in cancer rates is a pretty simple statistical phenomenon. Many years ago, there were lots of deaths which were caused by cancer, but we had no way to identify the cause as such. Instead, it might have been "consumption" (metabolic wasting) -- or maybe a stroke (actually caused by leukemia cells gumming up the bloodstream) -- or maybe a heart attack (caused by blood clots due to cancer-related coagulopathy).

Plus, not so long ago in historical terms human life expectancy was shorter, leading to less overall exposure to cell damaging influences -- you die for other reasons long before you get cancer.

Then, in more recent times there's the information revolution -- which leads to much more efficient reporting of cancer deaths and much more efficient dissemination of that info to the public.

So I would guess that cancers occur at about the same rate they always have, If you think about how cancer works, it's part and parcel of getting old. Young people can get cancer of course, but statistically you're much more likely to get it at a later age. The reasoning is as follows:

1) For every given tissue in the body, the cells of that tissue have a pre-programmed rate at which they divide/multiply, and a MATCHING pre-programmed rate at which they will die off (known as apoptosis).

2) Ergo, the number of cells in that tissue will remain approximately the same as long as the rates match -- which means the tissue will hold its shape/form/function whilst being continuously recycled and refreshed. This is why your body might have a lifespan of up to 100 years, even though your cells' lifespan is measured in months or even days.

3) Cancers occur when (in some cell, in some tissue, somewhere) the piece of DNA which governs the RATE gets damaged or simply obliterated. In that case the rate of cell replication is no longer governed, and the cell will multiply out of control at a rather fast rate. Or conversely, if the "apoptosis" DNA takes a hit, the cell will forget how/when to die off.

4) Either way, when the rates no longer match, you get a bunch of extra cells that shouldn't be there, and that's cancer.

5) The damaging force that nails the DNA can be, for example, radiation, chemicals, or even a virus (which has the capability to monkey with DNA). Or, in a few cases the DNA itself can be awry from the get-go -- such as some breast cancers where we are always looking for the predictive genetic markers.

6) Since with cancer we are talking about processes that recycle and refresh the body's tissues, that means we are talking about processes that are also integral to the aging process.

7) So, if we end up really curing cancer, that probably means we have figured out how to ADJUST the rate governance of cells/tissues to compensate for problems. If we can do THAT, not only have we fixed cancer at the root, we'll also have a leg up on tampering with aging. 500-year lifespans anyone? Think about the consequences of that before deciding whether it's a positive or negative outcome.

8) But we're a long way from discovering ourselves into that kind of ethical dilemma. In the meantime, we use chemotherapy, immunotherapy, or radiation to kill fast-replicating cells. It also means that the longer you live under the sun (which is one major source of cell damaging radiation!) the more chance there is that the rate DNA takes a hit somewhere in your body.