Looking for a data set that shows hospital patients' illnesses and their vital signs, during their stay I'm looking for a data set that shows hospital patients' vital signs (body temperature and/or heart rate, etc.) and their illnesses, over time. Is there such a data set?

For example:
Patient #1, Male, age:35, ..., Diagnosed w/: Heart failure
22:32:10: Temperature:32.1
22:32:20: Temperature:34.2
22:32:30: Temperature:33
22:32:40: Temperature:35

Or:
Patient #2, Female, age:35, ..., Diagnosed w/: Migraine Attack
22:32: Heart Rate:68
22:33: Heart Rate:75
22:34: Heart Rate:69
22:35: Heart Rate:70

I don't have any insistence for the data to come from a hospital. I already have searched some sources but with no luck.
I'm assuming you are looking for medical datasets, not only what you have stated in the question, but also some image recognition datasets which need something like CNN.

Try this one. This dataset contains several medical features including blood sugar, serum cholesterol etc, and wants you to find out the presence of heart disease.

In the meanwhile, there are some medical competitions and datasets on Kaggle, including the famous Data Science Bowl.