I think you can indeed think yourself ill just as you can speed your recovery through positive thinking. The hypochondriac might not end up suffering whatever disease they most fear, but can end up with something else. This can be made worse by the tendency of a hypochondriac to “coddle” themselves, thus not getting sunshine or exercise or all of the various things that people need to keep their immune systems strong.
The other difficulty is that if your doctor perceives you to be a hypochondriac, then he/she may discount your symptoms, thus letting an illness go on much longer without being checked.
This last happened to a friend of mine who felt unusually tired and run down, and who kept trying to tell her doctor about it over a period of 18 months. Her doctor kept putting her off with nonsense about her being a mom and so on. Finally, she cornered him and demanded he do something. He said he would do some tests to prove she was not ill. He came back a bit later and told her she had stage 4 lymphoma.