How do multiple PhD scholarships combine? I'm currently in the process of applying for various scholarships to fund my PhD. 

Since I'm writing to many different organizations/institutions/etc., I started wondering whether these different funding schemes can be combined. I found it hard to find any reliable information on this matter, so I reckoned I'd ask it here: 

Suppose I successfully obtain two scholarships, in two different countries, amounting to roughly $15.000 and $25.000 annually, both excluding coverage for university tuition fees. Should I notify both of them of having obtained the other? Or should I make absolutely sure they'll never find out :) ?

I suppose there'll be severe consequences to keeping it silent -- if so, what?
Dave's answer is mostly correct: you should not accept both without the explicit written permission of both scholarship providers.

Some scholarships can be combined, if the fellowship sponsor allows it. However, in general, large-scale fellowships cannot usually be accepted in conjunction with other fellowships. You'll need to choose one or the other. 

On the other hand, if you're talking about actual scholarships—by which I mean cash awards directly to you, without being tied to a specific course of study at a specific university—the rules may be more relaxed. 

The important thing is to read the rules and conditions associated with each award, and then to ask the sponsors if you have any questions.