I am a Biomedical Science graduate and it is a foregone conclusion that it is a combination of both.
Your genes are responsible for predispositions to certain illnesses, as well as susceptiblities and resistances to other infections and ailments. For instance, the european gene pool have an innate resistance to influenza, but vulnerabilities to less common illnesses in that population will remain.
In many cases, some people will be unable to influence the predisposition to certain illnesses is so strong there is little that can be done with treatment or any form of control, such as downs syndrome, fragile x syndrome, etc. There are also predispositions which can either not be influenced, or are weak enough so that an envrionmental trigger would be required, such a high intake of animal fats, depending on the strength of the predisposition, such as heart disease. Some people will find it hard to avoid, however, others will have a chance of avoiding it under the provision that they follow a stricter diet or take statins way before the onset of the condition.
Many of these genetic diseases require a “repeating unit” threshold to be higher than a certain level. For instance, you would need in your DNA a sequence to repeat over 50 times in order for you to have the condition. If your parents have, say, 30 each, you could inherit more than 30 from them, resulting in you having the disease, with them not showing it.
There is then the dominance of said genes causing such diseases. If something is dominant, having just 1 copy would result in you exhibiting the symptoms of the disease, without much envrionmental factors can do to influence it. If a disease is recessive, you would require both copies in your chromasomes to exhibit the disease. However, if you carry only 1, the disease would be suppressed, or alternately, only mildly exhibited, for example, in sickle cell anaemia. Alternately, some treatment and diets can help people cope with these infections, which easily ameliorate the symptoms, further lending to the concept that both genes and envrionment can influence many diseases.
As you can tell, there are many ways genes can affect diseases as well as help/hinder us fighting off infections. In immune flaws, people can be rendered totally unable to fight off typically simple infections, such as in the story of dede the tree man. He is unable to control a simple wart causing virus, resulting in his body being consumed by these normally, mildly irritating viral infections.
I could go on all day, however, I think this is enough :)