What circumstances could justify getting an extension on graduate admission offer deadline? A large number of US schools I applied to have put me on their waitlist, and since I also applied to a few programs in other Europe that release their decisions later, I still need more time to make my decision. (Especially because I've been put on the waitlist for the best graduate school in the world (or at least US) in my field, so it would be painful to accept a different offer, and then receive an offer from them that I cannot accept!)

I started receiving offers just this week, which is only a few days to the April 15th deadline, and I have at most 2 weeks to decide. That being said, I understand that there are other people on the waitlist, and I think the grad schools may say it's not possible to get an extension. But another problem I have for making the decision, is that I'm in a super-stressful situation for my finals now, and with all grad school stuff, the psychological pressure is high and I really believe I might make a bad decision in this situation. 

Is that (the stress and pressure) a good reason to ask for at least a few days more time for making the decision (so I'll be done with my exams and have a couple of days to make decision in a more relaxed condition), or is that too personal to explain to grad school and not acceptable?
Most of your colleagues are going through stressful scenarios right now as well, finishing up exams, projects, term papers, and the other parts of their undergraduate careers as well. So it’s unlikely that you’d get an extension just because of deadlines. 

If you had extreme extenuating circumstances—personal illness, family issues, or other events—you could make a request that is more likely to be accepted. But in general, there’s not much you can do.