These are two separate questions in my opinion. Factors like reputation of your undergraduate university and past admit history, matter quite a bit for admission to Masters as against PhDs. Hence, I might only be qualified to answer this question for BITS.
Unfortunately, I do not see a lot of BITSians making it to MIT. Harvard, like most other Ivy League Universities, is not highly ranked for graduate engineering programmes. Hence, I do not see a lot of students taking that option either. Most electronics/computer science students end up pursuing their masters in a tech-savvy engineering university with a strong Computer Engineering programme, since that’s got a bit for both of those disciplines. For an admission to any good programme, you need one of these to provide an edge to your application:
1.) Good GPA
2.) A couple of years of work experience
3.) Good course/lab projects to show on your resume
I am not really aware of the eligibility criteria, if any, other than having an undergraduate diploma from a certified institution.
Getting admitted into a programme solely depends on how well can you justify your admission through your statement of purpose. Theoretically, it is possible to get into any programme, provided you have enough background in that discipline. But to be more realistic, you have to prove the background you claim, which means, it shall be unlikely to take giant strides across disciplines.