Does everyone become ill if bit by an infected tick?

Not necessarily. In the case of lime disease it is estimated that the tick must be feeding on the body for many hours if not a whole day before the germ is passed in the process. The bite of any tick or insect can be secondarily infected by germs planted within a scratch at the site.
No. Although ticks can be carriers for some really nasty diseases, many people who have tick bites suffer only from the bite itself.