Yes, absolutely, one can do their post-graduation from any college after graduating from NID.
The National Institute of Design, NID is one of the best design schools, that is located in Ahmedabad, India. The institute is an autonomous body but works under the Department of Industrial Policy and Promotion, Ministry of Commerce and Industry, Government of India.
Even, according to Business Week NID topped the list of top design schools in the world in the year 2010.
The National Institute of Design gives its students the opportunity to access to professional education programs at Bachelor's and Masters's Levels with five faculty streams and 20 diverse and wide design domains through its creativity, new technology, and top-notch faculty. Not only this, but NID has also established exchange programs with more than 55 overseas institutes.
One thing should be kept in mind only those candidates who cleared the NID DAT Prelims will be able to qualify for the next round, i.e. NID DAT Mains. If the candidate passes both exams then they are called up for admission.