JEE Main

First things first: Please get this straight that admission to NITs and IIITs is nowhere related to JEE Advanced score. Hence, if you are targeting NITs or IIITs for any of the engineering branch, do not fret about your JEE advanced score. If youre targeting IITs or ISM then you need to look out for JEE advanced score.
Admission to NITs and IIITs is based solely on your JEE Mains score. And, a minimum percentage of 75% is required in 12th boards.
Here is how it works: You will be admitted into an NIT through CSAB (Central Seat Allocation Board) on the basis of your JEE mains score. An All India Rank is given to the candidate based on his score and he/she will be asked to pick his/her choices among NITs and some other private universities. So, if you have a good enough rank in JEE Mains, you will have a high chance of getting into a good NIT.
Earlier, 40% of board marks and 60% of JEE Mains were taken into consideration and a net total was calculated by CSAB to give a rank to the candidate. It is not the case anymore. According to JEE Main 2017 notification, no weightage will be given to 12th marks. The new protocol will work from 2017 onward.
Please note that weightage of 12th boards is removed but you still need to get a minimum of 75% in 12th exam. So, for example, you have secured a pretty nice score of 280 in JEE Mains but only 60/5 in 12th boards, you will not be qualified to get into an NIT or IIIT. For SC/ST, the qualifying marks would be 65% in 12th examination.
Now, simply clearing JEE Mains is not enough. You need to get a minimum good score. These ballpark figures will give you an idea about the scores you need to get in JEE Mains to get into an NIT or IIIT:
A score of 260+ is a power score in JEE Mains which is required to get admission into an IIIT.
For NITs, if you are ready to take any branch that is offered to you, i.e., if you are only concerned with the name of the college then a score of 150 in JEE Mains is enough. On the other hand, if you want a high profile branch like CS then a 200+ score is a must.
Last but not the least, if you have any other queries related to admission into NIT/IIIT/IIT, do check out IIT JEE Discussion Forum AskIITians is a great platform to solve all your doubts related to admissions into a good engineering college. You can post your questions in the forum where a counselor or student will respond to your query. Since, it is an engineering/medical-only forum; the queries are addressed very descriptively.