The incidence of contralateral breast cancer is high and constant with age, around five per 1000 women who had a primary breast cancer. For other neoplasms, the pattern of incidence of second primary neoplasms with age is less known, particularly as for only a few neoplasms the site of origin is not totally removed, and hence remains at risk of a second primary. Using the dataset from the Cancer Registry of the Swiss Canton of Vaud, we show that the incidence of second neoplasms is constant with age also after oral and pharyngeal, colorectal cancers, cutaneous malignant melanoma (CMM) and basal cell carcinoma. The incidence of first primary oral and pharyngeal cancer increased 20-fold between age 30-39 and 70-89 years, whereas the incidence of second neoplasms did not increase with age. Rates of second colorectal cancer remained relatively constant with age, between 2.5 per 1000 at age 40-59 years and 3.8 per 1000 at 70 years and above. Likewise, for CMM, the age-specific incidence rates of second primary CMM did not vary, ranging between 1 and 2.5 per 1000 in various subsequent age groups. The pattern of incidence for second basal cell carcinoma was similar, with no clear rise with age. These patterns are compatible with the occurrence of a single mutational event in a population of susceptible individuals. A possible implication of these observations is that a variable, but potentially large, proportion of cancers arise in very high-risk individuals and the incidence, on average, increases at a high constant level at a predetermined age.