please help, Question about the H1-B type ?

Consider the following situation.

If I work in one company for 6 years with H1B that is supported by the company, then I find another job in university (e.g. posdoc).

Since employ based H1B and non profit type H1B are different category, can I get another 6 years of non profit H1B (to work in college) ?

Thank you very much.