Hello Shreyas, Yeah, it may happen , but it is very rare . But honestly speaking you can get same date for some exams. From my personal experience, i can tell you that i got the same dates for ISI entrance exam and COMEDK. But that is very rare. I hope you don't suffer with the same.

All the best for your future endeavours. 

Hello Shreyansh

In most of the exams you get a chance to select your date of examination on your own. They give you options so that there's no problem for you in the future. However, few times ypur dates can go same. I think in those times it's those times you should work according to your priority and go for that exam.