Radiation and chemotherapy are both aggressive yet acceptable treatments for many forms of cancer. Some cancer patients receive both forms of treatment, but not usually at the same time, because the side effects of both treatments can be quite severe and weaken the patient (thereby putting their health at further risk). It is possible that someone with radiation poisoning might experience some effects that would mitigate cancer in the body, but it's very hard to say without doing a complete examination and having the opinion of an oncologist or other medical expert. Radiation sickness is effectively what results in many cancer patients receiving radiation treatments. So the logic posed in this question does have some practical value.
Chemotherapy often causes severe side effects in cancer patients; these side effects vary, but can include any of the following: hair loss, nausea, vomiting, diarrhea, loss of appetite, weakness, fatigue, body aches, loss of calcium in the bones, skin lesions, joint pain, muscle pain, headaches, and neuropathy, among others. Only a skilled oncologist should make a decision involving treatment. In the case of a cancer patient suffering radiation poisoning, the relevant risks would have to be measured against any potential benefit from the treatment.