My guess is no because there are many different kinds of cancer so each would require different conditions for it to develop.
Since my mother (47) and her mother (55) died of breast cancer, we made sure my daughter was getting routine screenings earlier than we would have otherwise. In my mother's case, they said she probably had it for 10 years prior to diagnosis and she was diagnosed at 45. That's just one example.
Now if everyone in the family had diets that were high nitrites or something and family recipes stayed around through generations or there was a family dynamic that everyone smoked.... The idea being that they are sharing one or more risk factors that are known risks for different types of cancer. Then they (as a group) would be more likely to get different types of cancers simply because they participate in high risk (for cancer) activities.
As far as I know, there is not a genetic condition that would span multiple types of cancer in a family tree. Environmental risk factors, can be more common in a family but it's exposure to the risk factor and not in the genes.
Again, just my opinion...
Remember, everybody used to just die of old age. Now, it's cancer or heart failure of some sort.
My father died of cancer, parents died of cancer (F-Lung, M-Ovarian).
My mother died of cancer, her parents died of cancer (F-Lung, M-Breast).
Mike