Yes obviously!! There are many students who have cracked the Neet Exam along with the BSc nursing course. It totally depends on you and your level of preparations.
Moreover you need not to worry about your BSc nursing exams. It is so because most of the subjects of BSC nursing in first year are same as what you studied for Neet Exam. From 2nd year onwards, the syllabus is new for you and you have to focus on BSc Nursing also.
So, best wishes to you if you want to prepare for NEET Exam. Hope you get your dreams achieved.